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[[File:Binary logarithm plot with ticks.svg|right|thumb|upright=1.35|alt=Graph showing a logarithm curves, which crosses the ''x''-axis where ''x'' is 1 and extend towards minus infinity along the ''y''-axis.|The [[graph of a function|graph]] of the logarithm to base 2 crosses the [[x axis|''x'' axis]] (horizontal axis) at 1 and passes through the points with [[coordinate]]s {{nowrap|(2, 1)}}, {{nowrap|(4, 2)}}, and {{nowrap|(8, 3)}}. For example, {{nowrap|log<sub>2</sub>(8) {{=}} 3}}, because {{nowrap|2<sup>3</sup> {{=}} 8.}} The graph gets arbitrarily close to the ''y'' axis, but [[asymptotic|does not meet or intersect it]].]]
[[Image:Logarithms.png|thumb|364px|Logarithms to various bases: <span style="color:red">red</span> is to base [[E (mathematical constant)|''e'']], <span style="color:green">green</span> is to base 10, and <span style="color:purple">purple</span> is to base 1.7. Each tick on the axis is one unit. Note how logarithms of all bases pass through the point (1, 0), because any nonzero number raised to the power 0 is 1, and through the points (''b'', 1) for base ''b'', because any number raised to the power 1 is itself.]]
 
   
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The '''logarithm''' of a number is the [[exponent]] by which another fixed value, the [[base (exponentiation)|base]], has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: {{nowrap|1000 {{=}} 10&thinsp;×&thinsp;10&thinsp;×&thinsp;10 {{=}} 10<sup>3</sup>.}} More generally, if {{nowrap begin}}''x'' = ''b''<sup>''y''</sup>{{nowrap end}}, then ''y'' is the logarithm of ''x'' to base&nbsp;''b'', and is written ''y'' = log<sub>''b''</sub>(''x''), so {{nowrap begin}}log<sub>10</sub>(1000) = 3.{{nowrap end}}
The '''logarithm''' is the [[mathematics|mathematical]] operation that is the [[inverse function|inverse]] of [[exponentiation]] (raising a constant, the ''base'', to a power).
 
   
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Logarithms were introduced by [[John Napier]] in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using [[slide rule]]s and [[Mathematical table|logarithm tables]]. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact&mdash;important in its own right&mdash;that the logarithm of a [[product (mathematics)|product]] is the [[sum]] of the logarithms of the factors:
The logarithm of a number ''x'' in base ''b'' is the number ''n'' such that ''x''&nbsp;= ''b''<sup>''n''</sup>. Thus ''b'' may never be 0 or a [[root (mathematics)|root]] of 1. It is usually written as
 
: <math> \log_b(x) = n \,</math>.
+
:<math> \log_b(xy) = \log_b (x) + \log_b (y). \,</math>
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The present-day notion of logarithms comes from [[Leonhard Euler]], who connected them to the [[exponential function]] in the 18th century.
   
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The logarithm to base {{nowrap begin}}''b'' = 10{{nowrap end}} is called the [[common logarithm]] and has many applications in science and engineering. The [[natural logarithm]] has the [[e (mathematical constant)|constant {{nowrap begin}}''e'']] (≈ 2.718{{nowrap end}}) as its base; its use is widespread in [[pure mathematics]], especially [[calculus]]. The [[binary logarithm]] uses base {{nowrap begin}}''b'' = 2{{nowrap end}} and is prominent in [[computer science]].
For example,
 
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:<math> \log_3(81) = 4 \,</math>
 
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[[Logarithmic scale]]s reduce wide-ranging quantities to smaller scopes. For example, the [[decibel]] is a logarithmic unit quantifying [[sound pressure]] and voltage ratios. In chemistry, [[pH]] and pOH are logarithmic measures for the [[acid]]ity of an [[aqueous solution]]. Logarithms are commonplace in scientific [[formula]]e, and in measurements of the [[Computational complexity theory|complexity of algorithms]] and of geometric objects called [[fractal]]s. They describe [[Interval (music)|musical intervals]], appear in formulae counting [[prime number]]s, inform some models in [[psychophysics]], and can aid in [[forensic accounting]].
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In the same way as the logarithm reverses [[exponentiation]], the [[complex logarithm]] is the [[inverse function]] of the exponential function applied to [[complex numbers]]. The [[discrete logarithm]] is another variant; it has applications in [[public-key cryptography]].
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==Motivation and definition==
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The idea of logarithms is to reverse the operation of [[exponentiation]], that is raising a number to a power. For example, the third power (or [[cube (algebra)|cube]]) of 2 is 8, because 8 is the product of three factors of 2:
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:<math>2^3 = 2 \times 2 \times 2 = 8. \,</math>
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It follows that the logarithm of 8 with respect to base 2 is 3, so log<sub>2</sub>&nbsp;8&nbsp;=&nbsp;3.
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===Exponentiation===
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The third power of some number ''b'' is the product of three factors of ''b''. More generally, raising ''b'' to the {{nowrap|''n''-th}} power, where ''n'' is a [[natural number]], is done by multiplying ''n'' factors of ''b''. The {{nowrap|''n''-th}} power of ''b'' is written ''b''<sup>''n''</sup>, so that
  +
:<math>b^n = \underbrace{b \times b \times \cdots \times b}_{n \text{ factors}}.</math>
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Exponentation may be extended to ''b''<sup>''y''</sup>, where ''b'' is a positive number and the ''exponent'' ''y'' is any [[real number]]. For example, ''b''<sup>&minus;1</sup> is the [[Multiplicative inverse|reciprocal]] of ''b'', that is, {{nowrap|1/''b''}}.{{#tag:ref|For further details, including the formula {{nowrap|''b''<sup>''m'' + ''n''</sup> <nowiki>=</nowiki> ''b''<sup>''m''</sup> &middot; ''b''<sup>''n''</sup>}}, see [[exponentiation]] or <ref>{{Citation|last1=Shirali| first1=Shailesh|title=A Primer on Logarithms|publisher=Universities Press|isbn=978-81-7371-414-6|year=2002|location=Hyderabad|url=http://books.google.com/books?id=0b0igbb3WaQC&printsec=frontcover#v=onepage&q&f=false}}, esp. section 2</ref> for an elementary treatise.|group=nb}}
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===Definition===
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The ''logarithm'' of a number ''x'' with respect to base ''b'' is the exponent by which ''b'' has to be raised to yield ''x''. In other words, the logarithm of ''x'' to base ''b'' is the solution ''y'' to the equation<ref>{{Citation|last1=Kate|first1=S.K.|last2=Bhapkar|first2=H.R.|title=Basics Of Mathematics|location=Pune|publisher=Technical Publications|isbn=978-81-8431-755-8|year=2009|url=http://books.google.com/books?id=v4R0GSJtEQ4C&pg=PR1#v=onepage&q&f=false}}, chapter 1</ref>
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: <math>b^y = x. \, </math>
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The logarithm is denoted "log<sub>''b''</sub>(''x'')" (pronounced as "the logarithm of ''x'' to base ''b''" or "the {{nowrap|base-''b''}} logarithm of ''x''"). In the equation ''y'' = log<sub>''b''</sub>(''x''), the value ''y'' is the answer to the question "To what power must ''b'' be raised, in order to yield ''x''?". For the logarithm to be defined, the base ''b'' must be a [[positive number|positive]] real number not equal to 1 and ''x'' must be a positive number.{{#tag:ref|The restrictions on ''x'' and ''b'' are explained in the section [[#Analytic properties|"Analytic properties"]].|group=nb}}
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===Examples===
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For example, {{nowrap|log<sub>2</sub>(16) {{=}} 4}}, since {{nowrap|2<sup>4</sup> {{=}} 2&thinsp;×2&thinsp;×&thinsp;2&thinsp;×&thinsp;2}} {{=}} 16. Logarithms can also be negative:
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:<math>\log_2 \!\left( \frac{1}{2} \right) = -1,\, </math>
 
since
 
since
:<math>3^4 = 3 \times 3 \times 3 \times 3 = 81 \,</math>.
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: <math>2^{-1} = \frac 1 {2^1} = \frac 1 2.</math>
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A third example: log<sub>10</sub>(150) is approximately 2.176, which lies between 2 and 3, just as 150 lies between {{nowrap|10<sup>2</sup> {{=}} 100}} and {{nowrap|10<sup>3</sup> {{=}} 1000}}. Finally, for any base ''b'', {{nowrap|log<sub>''b''</sub>(''b'') {{=}} 1}} and {{nowrap|1=log<sub>''b''</sub>(1) = 0}}, since {{nowrap|''b''<sup>1</sup> {{=}} ''b''}} and {{nowrap|''b''<sup>0</sup> {{=}} 1}}, respectively.
   
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==Logarithmic identities==
If ''n'' is a [[negative and non-negative numbers|positive]] [[integer]], ''b''<sup>''n''</sup> means the product of ''n'' factors equal to ''b''.
 
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{{Main|List of logarithmic identities}}
   
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Several important formulas, sometimes called ''logarithmic identities'' or ''log laws'', relate logarithms to one another.<ref>All statements in this section can be found in {{Harvard citations|last1=Shirali|first1=Shailesh|year=2002|loc=section 4|nb=yes}}, {{Harvard citations|last1=Downing| first1=Douglas |year=2003|loc=p. 275}}, or {{Harvard citations|last1=Kate|last2=Bhapkar|year=2009|loc=p. 1-1|nb=yes}}, for example.</ref>
:<math>\underbrace{b \times b \times \cdots \times b}_n</math>
 
   
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===Product, quotient, power, and root===
However, if ''b'' is a positive real number not equal to 1, the definition can be extended to any [[real number]] ''n'' in a [[field (mathematics)|field]] (at least a [[ring (mathematics)|ring]] with 1) (see [[exponentiation]] for details). Similarly, the logarithm function can be defined for any positive real number. For each positive base, ''b'', other than 1, there is one logarithm [[function (mathematics)|function]] and one exponential function; they are [[inverse function]]s. See the figure on the right.
 
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The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. Therefore, the logarithm of the {{nowrap|''p''-th}} power of a number is ''p'' times the logarithm of the number itself; the logarithm of a {{nowrap|''p''-th}} root is the logarithm of the number divided by ''p''. The following table lists these identities with examples:
   
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<center>
Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical operations easier to perform and, before the advent of [[electronic computer]]s, they were widely used for this purpose in fields such as [[astronomy]], [[engineering]], [[navigation]], and [[cartography]]. They have important mathematical properties and are still used in many ways.
 
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{| class="wikitable"
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|-
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! !! Formula !! Example
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|-
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| product || <cite id=labelLogarithmProducts><math> \log_b(x y) = \log_b (x) + \log_b (y) \,</math></cite>|| <math> \log_3 (243) = \log_3(9 \cdot 27) = \log_3 (9) + \log_3 (27) = 2 + 3 = 5 \,</math>
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|-
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| quotient || <math>\log_b \!\left(\frac x y \right) = \log_b (x) - \log_b (y) \,</math>|| <math> \log_2 (16) = \log_2 \!\left ( \frac{64}{4} \right ) = \log_2 (64) - \log_2 (4) = 6 - 2 = 4</math>
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|-
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| power || <cite id=labelLogarithmPowers><math>\log_b(x^p) = p \log_b (x) \,</math></cite>|| <math> \log_2 (64) = \log_2 (2^6) = 6 \log_2 (2) = 6 \,</math>
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|-
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| root || <math>\log_b \sqrt[p]{x} = \frac {\log_b (x)} p \, </math>|| <math> \log_{10} \sqrt{1000} = \frac{1}{2}\log_{10} 1000 = \frac{3}{2} = 1.5 </math>
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|}
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</center>
   
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===Change of base===<!-- This section is linked from [[Mathematica]] -->
==Bases==
 
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The logarithm log<sub>''b''</sub>(''x'') can be computed from the logarithms of ''x'' and ''b'' with respect to an arbitrary base ''k'' using the following formula:
The most widely used bases for logarithms are 10, the mathematical constant ''[[e (mathematical constant)|e]]'' ≈ 2.71828... and 2. When "log" is written without a base (''b'' missing from log<sub>''b''</sub>), the intent can usually be determined from context:
 
  +
: <cite id=labelLogarithmBaseChange><math> \log_b(x) = \frac{\log_k(x)}{\log_k(b)}.\, </math></cite>
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Typical [[scientific calculators]] calculate the logarithms to bases 10 and [[e (mathematical constant)|''e'']].<ref>{{Citation | last1=Bernstein | first1=Stephen | last2=Bernstein | first2=Ruth | title=Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability| publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-005023-5 | year=1999}}, p. 21</ref> Logarithms with respect to any base ''b'' can be determined using either of these two logarithms by the previous formula:
  +
:<math> \log_b (x) = \frac{\log_{10} (x)}{\log_{10} (b)} = \frac{\log_{e} (x)}{\log_{e} (b)}. \,</math>
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Given a number ''x'' and its logarithm log<sub>''b''</sub>(''x'') to an unknown base ''b'', the base is given by:
  +
: <math> b = x^\frac{1}{\log_b(x)}.</math>
   
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==Particular bases==
* [[natural logarithm]] (log<sub>''[[e (mathematical constant)|e]]</sub>'') in [[mathematical analysis]]
 
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Among all choices for the base ''b'', three are particularly common. These are ''b''&nbsp;=&nbsp;10, ''b''&nbsp;=&nbsp;[[e (mathematical constant)|''e'']] (the [[Irrational number|irrational]] mathematical constant ≈ 2.71828), and ''b''&nbsp;=&nbsp;2. In [[mathematical analysis]], the logarithm to base ''e'' is widespread because of its particular analytical properties explained below. On the other hand, {{nowrap|base-10}} logarithms are easy to use for manual calculations in the [[decimal]] number system:<ref>{{Citation|last1=Downing|first1=Douglas|title=Algebra the Easy Way|series=Barron's Educational Series|location=Hauppauge, N.Y.|publisher=Barron's|isbn=978-0-7641-1972-9|year=2003}}, chapter 17, p. 275</ref>
* [[common logarithm]] (log<sub>10</sub>) in [[engineering]] and when logarithm [[Mathematical table|table]]s are used to simplify hand calculations
 
  +
:<math>\log_{10}(10 x) = \log_{10}(10) + \log_{10}(x) = 1 + \log_{10}(x).\ </math>
* [[binary logarithm]] (log<sub>2</sub>) in [[information theory]] and [[interval (music)|musical interval]]s
 
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Thus, log<sub>10</sub>(''x'') is related to the number of [[decimal digit]]s of a positive integer ''x'': the number of digits is the smallest [[integer]] strictly bigger than log<sub>10</sub>(''x'').<ref>{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005}}, p. 20</ref> For example, log<sub>10</sub>(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in [[computer science]], where the [[binary numeral system|binary system]] is ubiquitous.
* [[indefinite logarithm]] when the base is irrelevant, e.g. in [[computational complexity theory|complexity theory]] when describing the asymptotic behavior of algorithms in [[big O notation]].
 
   
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The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(''x'') instead of log<sub>''b''</sub>(''x''), when the intended base can be determined from the context. The notation <sup>''b''</sup>log(''x'') also occurs.<ref>{{Citation| url=http://www.mathe-online.at/mathint/lexikon/l.html |author1=Franz Embacher |author2=Petra Oberhuemer |title=Mathematisches Lexikon |publisher=mathe online: für Schule, Fachhochschule, Universität unde Selbststudium |accessdate=22/03/2011 |language=German}}</ref> The "ISO notation" column lists designations suggested by the [[International Organization for Standardization]] ([[ISO 31-11]]).<ref>{{Citation| title = Guide for the Use of the International System of Units (SI)|author = B. N. Taylor|publisher = US Department of Commerce|year = 1995|url = http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2}}</ref>
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
 
   
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{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
===Other notations===
 
  +
|-
The notation "ln(''x'')" invariably means log<sub>e</sub>(''x''), i.e., the natural logarithm of ''x'', but the implied base for "log(''x'')" varies by discipline:
 
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! scope="col"|Base ''b''
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! scope="col"|Name for log<sub>''b''</sub>(''x'')
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! scope="col"|ISO notation
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! scope="col"|Other notations
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! scope="col"|Used in
  +
|-
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! scope="row"|2
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| [[binary logarithm]]
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| lb(''x'')<ref name=gullberg>{{Citation|title = Mathematics: from the birth of numbers.|author = Gullberg, Jan|location=New York|publisher = W. W. Norton & Co|year = 1997|isbn=978-0-393-04002-9}}</ref>
  +
| ld(''x''), log(''x''), lg(''x'')
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| computer science, [[information theory]], mathematics
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|-
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! scope="row"|''e''
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| [[natural logarithm]]
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| ln(''x''){{#tag:ref|Some mathematicians disapprove of this notation. In his 1985 autobiography, [[Paul Halmos]] criticized what he considered the "childish ln notation," which he said no mathematician had ever used.<ref>
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{{Citation
  +
|title = I Want to Be a Mathematician: An Automathography
  +
|author = Paul Halmos
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|publisher = Springer-Verlag
  +
|location=Berlin, New York
  +
|year = 1985
  +
|isbn=978-0-387-96078-4
  +
}}</ref>
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The notation was invented by [[Irving Stringham]], a mathematician.<ref>
  +
{{Citation
  +
|title = Uniplanar algebra: being part I of a propædeutic to the higher mathematical analysis
  +
|author = Irving Stringham
  +
|publisher = The Berkeley Press
  +
|year = 1893
  +
|page = xiii
  +
|url = http://books.google.com/?id=hPEKAQAAIAAJ&pg=PR13&dq=%22Irving+Stringham%22+In-natural-logarithm&q=
  +
}}</ref><ref>
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{{Citation|title = Introduction to Financial Technology|author = Roy S. Freedman|publisher = Academic Press|location=Amsterdam|year = 2006|isbn=978-0-12-370478-8|page = 59|url = http://books.google.com/?id=APJ7QeR_XPkC&pg=PA59&dq=%22Irving+Stringham%22+logarithm+ln&q=%22Irving%20Stringham%22%20logarithm%20ln
  +
}}</ref>|name=adaa|group=nb}}
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| log(''x'')<br>(in mathematics and many [[programming language]]s{{#tag:ref|For example [[C (programming language)|C]], [[Java (programming language)|Java]], [[Haskell (programming language)|Haskell]], and [[BASIC programming language|BASIC]].|group=nb}})
  +
| mathematical analysis, physics, chemistry,<br>[[statistics]], [[economics]], and some engineering fields
  +
|-
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! scope="row"|10
  +
| [[common logarithm]]
  +
| lg(''x'')
  +
| log(''x'')<br>(in engineering, biology, astronomy),
  +
| various [[engineering]] fields (see [[decibel]] and see below), <br>logarithm [[Mathematical table|tables]], handheld [[Scientific calculator|calculators]]
  +
|}
   
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==History==<!-- This section is linked from [[Common logarithm]] -->
* Mathematicians generally understand both "ln(''x'')" and "log(''x'')" to mean log<sub>e</sub>(''x'') and write "log<sub>10</sub>(''x'')" when the base-10 logarithm of ''x'' is intended.
 
   
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===Predecessors===
* Many engineers, biologists, astronomers, and some others write only "ln(''x'')" or "log<sub>e</sub>(''x'')" when they mean the natural logarithm of ''x'', and take "log(''x'')" to mean log<sub>10</sub>(''x'') or, sometimes in the context of [[computing]], [[binary logarithm|log<sub>2</sub>]](''x'').
 
  +
The [[Babylonian mathematics|Babylonians]] sometime in 2000–1600 BC may have invented the [[Multiplication algorithm#Quarter square multiplication|quarter square multiplication]] algorithm to multiply two numbers using only addition, subtraction and a table of squares.<ref>{{cite |title= Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers |last=McFarland |first=David |url=http://escholarship.org/uc/item/5n31064n |page=1 |year=2007}}</ref><ref>{{cite book| title=Mathematics in Ancient Iraq: A Social History |last=Robson |first=Eleanor | |page=227 |year=2008 |isbn= 978-0691091822 }}</ref> However it could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards till superseded by the use of computers.
   
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[[Michael Stifel]] published ''Arithmetica integra'' in [[Nuremberg]] in 1544 which contains a table<ref>{{Citation|first=Michaele|last=Stifelio|publisher=Iohan Petreium|location=London|year=1544|title=Arithmetica Integra|url = http://books.google.com/books?id=fndPsRv08R0C&pg=RA1-PT419}}</ref> of integers and powers of 2 that has been considered an early version of a logarithmic table.<ref>
* On most calculators, the LOG button is log<sub>10</sub>(''x'') and LN is log<sub>e</sub>(''x'').
 
  +
{{springer | title=Arithmetic | id= A/a013260 | last=Bukhshtab | first=A.A. | last2=Pechaev | first2=V.I.}}</ref><ref>
  +
{{Citation|title = Precalculus mathematics|author = Vivian Shaw Groza and Susanne M. Shelley|publisher = Holt, Rinehart and Winston|location=New York|year=1972|isbn=978-0-03-077670-0|page = 182|url = http://books.google.com/?id=yM_lSq1eJv8C&pg=PA182&dq=%22arithmetica+integra%22+logarithm&q=stifel}}</ref>
   
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In the 16th and early 17th centuries an algorithm called [[prosthaphaeresis]] was used to approximate multiplication and division. This used the trigonometric identity
* However, in most commonly used computer [[programming language]]s, including [[C programming language|C]], [[C++]], [[Java programming language|Java]], <!--[[Pascal programming language|Pascal]], -->[[Fortran]], and [[BASIC programming language|BASIC]], the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."
 
  +
:<math>\cos\,\alpha\,\cos\,\beta = \frac12[\cos(\alpha+\beta) + \cos(\alpha-\beta)]</math>
  +
or similar to convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.
   
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===From Napier to Euler===
* Some people use Log(''x'') (capital ''L'') to mean log<sub>10</sub>(''x''), and use log(''x'') with a lowercase ''l'' to mean log<sub>''e''</sub>(''x'').
 
   
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[[File:John Napier.jpg|thumb|right|John Napier (1550–1617), the inventor of logarithms|alt=A baroque picture of a sitting man with a beard.]]
* The notation Log(''x'') is also used by mathematicians to denote the [[principal branch]] of the (natural) logarithm function.
 
   
  +
The method of logarithms was publicly propounded by [[John Napier]] in 1614, in a book entitled ''Mirifici Logarithmorum Canonis Descriptio'' (''Description of the Wonderful Rule of Logarithms'').<ref>{{Citation|author=Ernest William Hobson|title=John Napier and the invention of logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=http://www.archive.org/details/johnnapierinvent00hobsiala}}</ref> [[Joost Bürgi]] independently invented logarithms but published six years after Napier.<ref>{{Harvard citations
* Also frequently used is the notation <sup>''b''</sup>log(''x'') instead of log<sub>''b''</sub>(''x'').
 
  +
|last1=Boyer|year=1991 |nb=yes |loc=Chapter 14, section "Jobst Bürgi"}}</ref>
   
  +
[[Johannes Kepler]], who used logarithm tables extensively to compile his ''Ephemeris'' and therefore dedicated it to John Napier,<ref>{{Citation |title=John Napier: Logarithm John |first=Lynne |last=Gladstone-Millar |publisher=National Museums Of Scotland |year=2003 |isbn=978-1-901663-70-9}}, p. 44</ref> remarked:
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/''x'', and having a simple definition), while, back when logarithms were used to speed computations, the decimal logarithms were far better for that, so natural logarithms were unknown outside calculus classes while decimal logarithms were widely used in a variety of disciplines.
 
  +
{{quote|...the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ...instead of rearing up his child for the public benefit he deserted it in the birth.|Johannes Kepler<ref>{{Citation |last=Napier |first=Mark |authorlink=Mark Napier (historian) |title=Memoirs of John Napier of Merchiston |publisher=William Blackwood |location=Edinburgh |year=1834 |url=http://books.google.com/books?id=husGAAAAYAAJ&pg=PA1&source=gbs_toc_r&cad=4#v=onepage&q&f=false}}, p. 392.</ref>|Rudolphine Tables (1627)}}
   
  +
By repeated subtractions Napier calculated {{nowrap|(1 − 10<sup>−7</sup>)<sup>''L''</sup>}} for ''L'' ranging from 1 to 100. The result for ''L''=100 is approximately {{nowrap begin}}0.99999 = 1 − 10<sup>−5</sup>{{nowrap end}}. Napier then calculated the products of these numbers with {{nowrap|10<sup>7</sup>(1 − 10<sup>−5</sup>)<sup>''L''</sup>}} for ''L'' from 1 to 50, and did similarly with {{nowrap|0.9998 ≈ (1 − 10<sup>−5</sup>)<sup>20</sup>}} and {{nowrap|0.9 ≈ 0.995<sup>20</sup>}}. These computations, which occupied 20 years, allowed him to give, for any number ''N'' from 5 to 10 million, the number ''L'' that solves the equation
As recently as 1984, [[Paul Halmos]] in his "automathography" ''I Want to Be a Mathematician'' heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at [[University of California, Berkeley|Berkeley]].) [[As of 2005]], many mathematicians have adopted the "ln" notation, but most use "log".
 
   
  +
:<math>N=10^7 {(1-10^{-7})}^L. \,</math>
In computer science, the base 2 logarithm is sometimes written as lg(''x'') to avoid confusion. This usage was suggested by Edward Reingold and popularized by [[Donald Knuth]]. However, in Russian literature, the notation lg(''x'') is generally used for the base 10 logarithm, so even this usage is not without its perils.<ref>[http://mathworld.wolfram.com/CommonLogarithm.html "Common Logarithm" at MathWorld]</ref>.
 
   
  +
Napier first called ''L'' an "artificial number", but later introduced the word ''"logarithm"'' to mean a number that indicates a ratio: {{lang|grc|λόγος}} (''[[logos]]'') meaning proportion, and {{lang|grc|ἀριθμός}} (''arithmos'') meaning number. In modern notation, the relation to natural logarithms is:
=== Change of base ===
 
  +
<ref>{{Citation
While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually log<sub>''e''</sub> and log<sub>10</sub>). To find a logarithm with base ''b'', using any other base ''k'':
 
  +
| title = The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature ; the R.S. Peale reprint,
  +
| volume = 17
  +
| edition = 9th
  +
| author = William Harrison De Puy
  +
| publisher = Werner Co.
  +
| year = 1893
  +
| page = 179
  +
| url = http://babel.hathitrust.org/cgi/pt?seq=7&view=image&size=100&id=nyp.33433082033444&u=1&num=179
  +
}}</ref>
   
  +
:<math>L = \log_{(1-10^{-7})} \!\left( \frac{N}{10^7} \right) \approx 10^7 \log_{ \frac{1}{e}} \!\left( \frac{N}{10^7} \right) = -10^7 \log_e \!\left( \frac{N}{10^7} \right),</math>
: <math> \log_b(x) = \frac{\log_k(x)}{\log_k(b)}. </math>
 
   
  +
where the very close approximation corresponds to the observation that
Moreover, this result implies that all logarithm functions (whatever the base) are [[similar]] to each other. So to calculate the log with base 2 of the number 16 with your calculator:
 
   
: <math> \log_2(16) = \frac{\log(16)}{\log(2)}. </math>
+
:<math>{(1-10^{-7})}^{10^7} \approx \frac{1}{e}. \,</math>
   
  +
The invention was quickly and widely met with acclaim. The works of [[Bonaventura Cavalieri]] (Italy), [[Edmund Wingate]] (France), Xue Fengzuo (China), and
== Uses of logarithms ==
 
  +
[[Johannes Kepler]]'s ''Chilias logarithmorum'' (Germany) helped spread the concept further.<ref>
Logarithms are useful in solving equations in which exponents are unknown. They have simple [[derivative]]s, so they are often used in the solution of [[integral]]s. The logarithm is one of three closely related functions. In the equation ''b''<sup>''n''</sup> = ''x'', ''b'' can be determined with [[radical (mathematics)|radical]]s, ''n'' with logarithms, and ''x'' with [[exponential function|exponentials]]. See [[logarithmic identities]] for several rules governing the logarithm functions. For a discussion of some additional aspects of logarithms see [[additional logarithm topics]].
 
  +
{{Citation|last1=Maor|first1=Eli|title=e: The Story of a Number|publisher=[[Princeton University Press]]|isbn=978-0-691-14134-3|year=2009}}, section 2</ref>
   
  +
[[File:1 over x integral.svg|The hyperbola {{nowrap|''y'' {{=}} 1/''x''}} (red curve) and the area from ''x'' = 1 to 6 (shaded in orange).|right|thumb]]
===Science and engineering===
 
  +
In 1647 [[Grégoire de Saint-Vincent]] related logarithms to the quadrature of the hyperbola, by pointing out that the area ''f''(''t'') under the hyperbola from {{nowrap|''x'' {{=}} 1}} to {{nowrap|''x'' {{=}} ''t''}} satisfies
Various quantities in science are expressed as logarithms of other quantities; see [[logarithmic scale]] for an explanation and a more complete list.
 
  +
:<math>f(tu) = f(t) + f(u).\,</math>
  +
The natural logarithm was first described by [[Nicholas Mercator]] in his work ''Logarithmotechnia'' published in 1668,<ref>{{Citation|author1=J. J. O'Connor|author2=E. F. Robertson |url=http://www-history.mcs.st-and.ac.uk/HistTopics/e.html |title=The number e |publisher=The MacTutor History of Mathematics archive |date=2001-09 |accessdate=02/02/2009}}</ref> although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm.<ref>{{Citation|last=Cajori |first=Florian |authorlink=Florian Cajori |title=A History of Mathematics|edition=5th|location=Providence, RI|publisher=AMS Bookstore |year=1991 |isbn=978-0-8218-2102-2|url=http://books.google.com/?id=mGJRjIC9fZgC&printsec=frontcover#v=onepage&q=speidell&f=false}}, p. 152</ref> Around 1730, [[Leonhard Euler]] defined the exponential function and the natural logarithm by
  +
:<math>e^x = \lim_{n \rightarrow \infty} (1+x/n)^n,</math>
  +
:<math>\ln(x) = \lim_{n \rightarrow \infty} n(x^{1/n} - 1).</math>
  +
Euler also showed that the two functions are inverse to one another.<ref name="ReferenceA">
  +
{{Harvard citations
  +
|last1=Maor |year=2009 |nb=yes |loc=sections 1, 13}}</ref><ref>{{Citation |last1=Eves |first1=Howard Whitley |author1-link=Howard Eves |title=An introduction to the history of mathematics |publisher=Saunders |location=Philadelphia |edition=6th |series=The Saunders series |isbn=978-0-03-029558-4 |year=1992}}, section 9-3</ref><ref>{{Citation | last1=Boyer | first1=Carl B. | author1-link=Carl Benjamin Boyer | title=A History of Mathematics | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-54397-8 | year=1991}}, p. 484, 489</ref>
   
  +
===Logarithm tables, slide rules, and historical applications{{anchor|Antilogarithm}}===
*The negative of the base-10 logarithm is used in [[chemistry]], where it expresses the [[concentration]] of hydronium ions ([[pH]]). The concentration of hydronium ions in neutral [[water]] is 10<sup>&minus;7</sup> at 25 °C, hence a pH of 7.
 
   
  +
[[Image:Logarithms Britannica 1797.png|thumb|360px|right|The 1797 ''[[Encyclopædia Britannica]]'' explanation of logarithms]]
*The ''bel'' (symbol B) is a [[Units of measurement|unit]] of measure which is the base-10 logarithm of [[ratio]]s, such as [[power (physics)|power]] levels and [[voltage]] levels. It is mostly used in [[telecommunication]], [[electronics]], and [[acoustics]]. It is used, in part, because the ear responds logarithmically to acoustic power. The Bel is named after telecommunications pioneer [[Alexander Graham Bell]]. The ''[[decibel]]'' (dB), equal to 0.1&nbsp;bel, is more commonly used. The ''[[neper]]'' is a similar unit which uses the natural logarithm of a ratio.
 
   
  +
By simplifying difficult calculations, logarithms contributed to the advance of science, and especially of [[astronomy]]. They were critical to advances in [[surveying]], [[celestial navigation]], and other domains. [[Pierre-Simon Laplace]] called logarithms
*The [[Richter scale]] measures earthquake intensity on a base-10 logarithmic scale.
 
   
  +
<blockquote>
*In spectrometry and optics, the absorbance unit used to measure [[optical density]] is equivalent to −1&nbsp;B.
 
  +
<p>[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.<ref>
  +
{{Citation |last1=Bryant |first1=Walter W. |title=A History of Astronomy |url=http://www.forgottenbooks.org/ebooks/A_History_of_Astronomy_-_9781440057922.pdf |publisher=Methuen & Co|location=London }}, p. 44</ref></p>
  +
</blockquote>
   
  +
A key tool that enabled the practical use of logarithms before calculators and computers was the ''table of logarithms''.<ref>{{Citation | last1=Campbell-Kelly | first1=Martin | title=The history of mathematical tables: from Sumer to spreadsheets | publisher=[[Oxford University Press]] | series=Oxford scholarship online | isbn=978-0-19-850841-0 | year=2003}}, section 2</ref> The first such table was compiled by [[Henry Briggs (mathematician)|Henry Briggs]] in 1617, immediately after Napier's invention. Subsequently, tables with increasing scope and precision were written. These tables listed the values of log<sub>''b''</sub>(''x'') and ''b''<sup>''x''</sup> for any number ''x'' in a certain range, at a certain precision, for a certain base ''b'' (usually {{nowrap begin}}''b'' = 10{{nowrap end}}). For example, Briggs' first table contained the common logarithms of all integers in the range 1&ndash;1000, with a precision of 8 digits. As the function {{nowrap|''f''(''x'') {{=}} ''b''<sup>''x''</sup>}} is the inverse function of log<sub>''b''</sub>(''x''), it has been called the antilogarithm.<ref>{{Citation|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun|title=[[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]|publisher=[[Dover Publications]]|location=New York|isbn=978-0-486-61272-0|edition=10th|year=1972}}, section 4.7., p. 89</ref> The product and quotient of two positive numbers ''c'' and ''d'' were routinely calculated as the sum and difference of their logarithms. The product ''cd'' or quotient ''c''/''d'' came from looking up the antilogarithm of the sum or difference, also via the same table:
*In astronomy, the [[apparent magnitude]] measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
 
  +
:<math> c d = b^{\log_b (c)} \, b^{\log_b (d)} = b^{\log_b (c) + \log_b (d)} \,</math>
  +
and
  +
:<math>\frac c d = c d^{-1} = b^{\log_b (c) - \log_b (d)}. \,</math>
   
  +
For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as [[prosthaphaeresis]], which relies on [[trigonometric identities]]. Calculations of powers and [[nth root|roots]] are reduced to multiplications or divisions and look-ups by
=== Exponential functions ===
 
  +
:<math>c^d = (b^{\log_b (c) })^d = b^{d \log_b (c)} \,</math>
The natural exponential function exp(x), also written <math>e^x</math> is defined as the inverse of the natural logarithm. It is positive for every real argument x.
 
  +
and
  +
:<math>\sqrt[d]{c} = c^{\frac 1 d} = b^{\frac{1}{d} \log_b (c)}. \,</math>
   
  +
Many logarithm tables give logarithms by separately providing the characteristic and [[common logarithm|mantissa]] of ''x'', that is to say, the [[integer part]] and the [[fractional part]] of log<sub>10</sub>(''x'').<ref>{{Citation | last1=Spiegel | first1=Murray R. | last2=Moyer | first2=R.E. | title=Schaum's outline of college algebra | publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-145227-4 | year=2006}}, p. 264</ref> The characteristic of {{nowrap|10 &middot; ''x''}} is one plus the characteristic of ''x'', and their [[significand]]s are the same. This extends the scope of logarithm tables: given a table listing log<sub>10</sub>(''x'') for all integers ''x'' ranging from 1 to 1000, the logarithm of 3542 is approximated by
The operation of "raising b to a power p" for positive arguments <math>b</math> and all real exponents <math>p</math> is defined by
 
  +
:<math>\log_{10}(3542) = \log_{10}(10\cdot 354.2) = 1 + \log_{10}(354.2) \approx 1 + \log_{10}(354). \, </math>
   
  +
Another critical application was the [[slide rule]], a pair of logarithmically divided scales used for calculation, as illustrated here:
:<math>\begin{matrix}b^p & = & \exp({p\ln b })\, \end{matrix}</math>
 
   
  +
[[Image:Slide rule example2 with labels.svg|center|thumb|550px|Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to ''x'' is proportional to the logarithm of ''x''.|alt=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.]]
   
  +
The non-sliding logarithmic scale, [[Gunter's rule]], was invented shortly after Napier's invention. [[William Oughtred]] enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.<ref name="ReferenceA"/>
The '''antilogarithm''' function is another name for the inverse of the logarithmic function. It is written antilog<sub>''b''</sub>(''n'') and means the same as <math>b^n</math>.
 
   
  +
==Analytic properties==
=== Easier computations ===
 
  +
A deeper study of logarithms requires the concept of a ''[[function (mathematics)|function]]''. A function is a rule that, given one number, produces another number.<ref>{{Citation | last1=Devlin | first1=Keith | author1-link=Keith Devlin | title=Sets, functions, and logic: an introduction to abstract mathematics | publisher=Chapman & Hall/CRC | location=Boca Raton, Fla | edition=3rd | series=Chapman & Hall/CRC mathematics | isbn=978-1-58488-449-1 | year=2004}}, or see the references in [[function (mathematics)|function]]</ref> An example is the function producing the {{nowrap|''x''-th}} power of ''b'' from any real number ''x'', where the base (or [[radix]]) ''b'' is a fixed number. This function is written
Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes certain operations easier:
 
  +
:<math>f(x) = b^x. \, </math>
{| style="border:1px solid" align="center"
 
|- align="center"
 
! Operation with numbers !! Operation with exponents !! Logarithmic identity
 
|- align="center"
 
| <math> \!\, a b </math> || <math> \!\, A + B </math> || <math> \!\, \log(a b) = \log(a) + \log(b) </math>
 
|- align="center"
 
| <math> \!\, a / b </math> || <math> \!\, A - B </math> || <math> \!\, \log(a / b) = \log(a) - \log(b) </math>
 
|- align="center"
 
| <math> \!\, a ^ b </math> || <math> \!\, A b </math> || <math> \!\, \log(a ^ b) = b \log(a) </math>
 
|- align="center"
 
| <math> \!\, \sqrt[b]{a} </math> || <math> \!\, A / b </math> || <math> \!\, \log(\sqrt[b]{a}) = \frac{\log(a)}{b} </math>
 
|}
 
These relations made such operations on two numbers much faster and the proper use of logarithms was an essential skill before multiplying [[calculator]]s became available.
 
   
  +
===Logarithmic function===
The <math>\log(a b) = \log(a) +\log(b)</math> equation is fundamentally (it implies effectively the other three relations in a field) because it
 
  +
To justify the definition of logarithms, it is necessary to show that the equation
describes an [[isomorphism]] between the '''additive group''' and the '''multiplicative group''' of the field.
 
  +
:<math>b^x = y \,</math>
  +
has a solution ''x'' and that this solution is unique, provided that ''y'' is positive and that ''b'' is positive and unequal to 1. A proof of that fact requires the [[intermediate value theorem]] from elementary [[calculus]].<ref name=LangIII.3>{{Citation|last1=Lang|first1=Serge|author1-link=Serge Lang|title=Undergraduate analysis|publisher=[[Springer-Verlag]]|location=Berlin, New York|edition=2nd|series=Undergraduate Texts in Mathematics|isbn=978-0-387-94841-6|mr=1476913|year=1997}}, section III.3</ref> This theorem states that a [[continuous function]] which produces two values ''m'' and ''n'' also produces any value that lies between ''m'' and ''n''. A function is ''continuous'' if it does not "jump", that is, if its graph can be drawn without lifting the pen.
   
  +
This property can be shown to hold for the function {{nowrap begin}}''f''(''x'') = ''b''<sup>''x''</sup>{{nowrap end}}. Because ''f'' takes arbitrarily large and arbitrarily small positive values, any number {{nowrap|''y'' > 0}} lies between ''f''(''x''<sub>0</sub>) and ''f''(''x''<sub>1</sub>) for suitable ''x''<sub>0</sub> and ''x''<sub>1</sub>. Hence, the intermediate value theorem ensures that the equation ''f''(''x'') = ''y'' has a solution. Moreover, there is only one solution to this equation, because the function ''f'' is [[monotonic function|strictly increasing]] (for {{nowrap|''b'' > 1}}), or strictly decreasing (for {{nowrap|0 < ''b'' < 1}}).<ref name=LangIV.2 />
To multiply two numbers, one found the logarithms of both numbers on a table of [[common logarithm]]s, added them, and then looked up the result in the table to find the product. This is faster than multiplying them by hand, provided that more than two decimal figures are needed in the result. The table needed to get an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few shelves.
 
   
  +
The unique solution ''x'' is the logarithm of ''y'' to base ''b'', log<sub>''b''</sub>(''y''). The function which assigns to ''y'' its logarithm is called ''logarithm function'' or ''logarithmic function'' (or just ''logarithm'').
The discovery of logarithms just before Newton's era had an impact in the scientific world which can be compared with the invention of the computer in the 20th century, because many calculations which were too laborious became feasible.
 
   
  +
===Inverse function===
When the chronometer was invented in the 18th century, logarithms allowed all calculations needed for astronomical navigation to be reduced to just additions, speeding the process by one or two orders of magnitude. A table of logarithms with five decimals, plus logarithms of trigonometric functions, was enough for most astronomical navigation calculations, and those tables fit in a small book.
 
  +
[[File:Logarithm inversefunctiontoexp.svg|right|thumb|The graph of the logarithm function log<sub>''b''</sub>(''x'') (blue) is obtained by [[Reflection (mathematics)|reflecting]] the graph of the function ''b''<sup>''x''</sup> (red) at the diagonal line ({{nowrap begin}}''x'' = ''y''{{nowrap end}}).|alt=The graphs of two functions.]]
  +
The formula for the logarithm of a power says in particular that for any number ''x'',
  +
:<math>\log_b \left (b^x \right) = x \log_b(b) = x.</math>
  +
In prose, taking the {{nowrap|''x''-th}} power of ''b'' and then the {{nowrap|base-''b''}} logarithm gives back ''x''. Conversely, given a positive number ''y'', the formula
  +
:<math>b^{\log_b(y)} = y</math>
  +
says that first taking the logarithm and then exponentiating gives back ''y''. Thus, the two possible ways of combining (or [[composition (mathematics)|composing]]) logarithms and exponentiation give back the original number. Therefore, the logarithm to base ''b'' is the ''[[inverse function]]'' of {{nowrap|''f''(''x'') {{=}} ''b''<sup>''x''</sup>}}.<ref>{{Citation | last1=Stewart | first1=James | title=Single Variable Calculus: Early Transcendentals | publisher=Thomson Brooks/Cole |location=Belmont|isbn=978-0-495-01169-9 | year=2007}}, section 1.6</ref>
   
  +
Inverse functions are closely related to the original functions. Their [[graph (mathematics)|graphs]] correspond to each other upon exchanging the ''x''- and the ''y''-coordinates (or upon reflection at the diagonal line ''x'' = ''y''), as shown at the right: a point (''t'', ''u'' = ''b''<sup>''t''</sup>) on the graph of ''f'' yields a point (''u'', ''t'' = log<sub>''b''</sub>''u'') on the graph of the logarithm and vice versa. As a consequence, log<sub>''b''</sub>(''x'') [[divergent sequence|diverges to infinity]] (gets bigger than any given number) if ''x'' grows to infinity, provided that ''b'' is greater than one. In that case, log<sub>''b''</sub>(''x'') is an [[increasing function]]. For {{nowrap|''b'' < 1}}, log<sub>''b''</sub>(''x'') tends to minus infinity instead. When ''x'' approaches zero, log<sub>''b''</sub>(''x'') goes to minus infinity for {{nowrap|''b'' > 1}} (plus infinity for {{nowrap|''b'' < 1}}, respectively).
To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix. [[Interpolation]] could be used for still higher precision. [[Slide rule]]s used logarithms to perform the same operations more rapidly, but with much less precision than using tables. Other tools for performing multiplications before the invention of the calculator include [[Napier's bones]] and mechanical calculators: see [[history of computing hardware]].
 
   
  +
===Derivative and antiderivative===
=== Calculus ===
 
The [[derivative]] of the natural logarithm function is:
+
[[File:Logarithm derivative.svg|right|thumb|220|The graph of the natural logarithm (green) and its tangent at {{nowrap|''x'' {{=}} 1.5}} (black)|alt=A graph of the logarithm function and a line touching it in one point.]]
  +
Analytic properties of functions pass to their inverses.<ref name=LangIII.3 /> Thus, as {{nowrap begin}}''f''(''x'') = ''b''<sup>''x''</sup>{{nowrap end}} is a continuous and [[differentiable function]], so is log<sub>''b''</sub>(''y''). Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the [[derivative]] of ''f''(''x'') evaluates to ln(''b'')''b''<sup>''x''</sup> by the properties of the [[exponential function]], the [[chain rule]] implies that the derivative of log<sub>''b''</sub>(''x'') is given by<ref name=LangIV.2>{{Harvard citations|last1=Lang|year=1997 |nb=yes|loc=section IV.2}}</ref><ref>{{cite web |work=Wolfram Alpha |title=Calculation of ''d/dx(Log(b,x))'' |publisher=[[Wolfram Research]] |accessdate=15 March 2011 |url=http://www.wolframalpha.com/input/?i=d/dx(Log(b,x)) }}</ref>
: <math>\frac{d}{dx} \ln(x) = \frac{1}{x}.</math>
 
  +
: <math>\frac{d}{dx} \log_b(x) = \frac{1}{x\ln(b)}. </math>
  +
That is, the [[slope]] of the [[tangent]] touching the graph of the {{nowrap|base-''b''}} logarithm at the point {{nowrap|(''x'', log<sub>''b''</sub>(''x''))}} equals {{nowrap|1/(''x''&thinsp;ln(''b''))}}. In particular, the derivative of ln(''x'') is 1/''x'', which implies that the [[antiderivative]] of 1/''x'' is {{nowrap|ln(''x'') + C}}. The derivative with a generalised functional argument ''f''(''x'') is
  +
:<math>\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}.</math>
  +
The quotient at the right hand side is called the [[logarithmic derivative]] of ''f''. Computing ''f<nowiki>'</nowiki>''(''x'') by means of the derivative of ln(''f''(''x'')) is known as [[logarithmic differentiation]].<ref>{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Calculus: an intuitive and physical approach | publisher=[[Dover Publications]] | location=New York | series=Dover books on mathematics | isbn=978-0-486-40453-0 | year=1998}}, p. 386</ref> The antiderivative of the natural logarithm ln(''x'') is:<ref>{{cite web |work=Wolfram Alpha |title=Calculation of ''Integrate(ln(x))'' |publisher=Wolfram Research |accessdate=15 March 2011 |url=http://www.wolframalpha.com/input/?i=Integrate(ln(x)) }}</ref>
  +
: <math>\int \ln(x) \,dx = x \ln(x) - x + C.</math>
  +
[[List of integrals of logarithmic functions|Related formulas]], such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<ref>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 69}}</ref>
   
  +
===Integral representation of the natural logarithm===
and by applying the change-of-base rule, the derivative for other bases is:
 
  +
[[File:Natural logarithm integral.svg|right|thumb|The natural logarithm of ''t'' is the shaded area underneath the graph of the function ''f''(''x'') = 1/''x'' (reciprocal of ''x'').|alt=A hyperbola with part of the area underneath shaded in grey.]]
  +
The natural logarithm of ''t'' agrees with the [[integral]] of 1/''x''&nbsp;''dx'' from 1 to ''t'':
  +
:<cite id=integral_naturallog><math>\ln (t) = \int_1^t \frac{1}{x} \, dx.</math></cite>
  +
In other words, ln(''t'') equals the area between the ''x'' axis and the graph of the function 1/''x'', ranging from {{nowrap begin}}''x'' = 1{{nowrap end}} to {{nowrap begin}}''x'' = ''t''{{nowrap end}} (figure at the right). This is a consequence of the [[fundamental theorem of calculus]] and the fact that derivative of ln(''x'') is 1/''x''. The right hand side of this equation can serve as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition.<ref>{{Citation|last1=Courant|first1=Richard|title=Differential and integral calculus. Vol. I|publisher=[[John Wiley & Sons]]|location=New York|series=Wiley Classics Library|isbn=978-0-471-60842-4|mr=1009558|year=1988}}, section III.6</ref> For example, the product formula {{nowrap begin}}ln(''tu'') = ln(''t'') + ln(''u''){{nowrap end}} is deduced as:
   
: <math>\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}</math>
+
:<math> \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw = \ln(t) + \ln(u).</math>
   
  +
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable ({{nowrap begin}}''w'' = ''x''/''t''{{nowrap end}}). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor ''t'' and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function {{nowrap begin}}''f''(''x'') = 1/''x''{{nowrap end}} again. Therefore, the left hand blue area, which is the integral of ''f''(''x'') from ''t'' to ''tu'' is the same as the integral from ''1'' to ''u''. This justifies the equality (2) with a more geometric proof.
The [[antiderivative]] of the logarithm is
 
: <math>\int \log_b(x) \,dx = x \log_b(x) - \frac{x}{\ln(b)} + C = x \log_b \left(\frac{x}{e}\right) + C.</math>
 
   
  +
[[File:Natural logarithm product formula proven geometrically.svg|thumb|center|500px|A visual proof of the product formula of the natural logarithm|alt=The hyperbola depicted twice. The area underneath is split into different parts.]]
''See also:'' [[Wikisource:Table of common limits#Logarithmic and exponential functions|table of limits of logarithmic functions]], [[list of integrals of logarithmic functions]].
 
   
  +
The power formula {{nowrap begin}}ln(''t''<sup>''r''</sup>) = ''r'' ln(''t''){{nowrap end}} may be derived in a similar way:
==Series for calculating the natural logarithm==
 
There are several series for calculating natural logarithms.<ref>''Handbook of Mathematical Functions'', National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68.</ref> The simpliest (but inefficient) is:
 
   
  +
:<math>
:<math>\ln (z) = \sum_{n=1}^\infty \frac{-{(-1)}^n}{n} (z-1)^n</math>
 
  +
\ln(t^r) = \int_1^{t^r} \frac{1}{x}dx = \int_1^t \frac{1}{w^r} \left(rw^{r - 1} \, dw\right) = r \int_1^t \frac{1}{w} \, dw = r \ln(t).
  +
</math>
  +
The second equality uses a change of variables ([[integration by substitution]]), {{nowrap begin}}''w'' := ''x''<sup>1/''r''</sup>{{nowrap end}}.
   
  +
The sum over the reciprocals of natural numbers,
when ''z'' is within 1 of 1.
 
  +
:<math>1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 n = \sum_{k=1}^n \frac{1}{k},</math>
  +
is called the [[harmonic series (mathematics)|harmonic series]]. It is closely tied to the natural logarithm: as ''n'' tends to [[infinity]], the difference,
  +
:<math>\sum_{k=1}^n \frac{1}{k} - \ln(n),</math>
  +
[[limit of a sequence|converges]] (i.e., gets arbitrarily close) to a number known as the [[Euler–Mascheroni constant]]. This relation aids in analyzing the performance of algorithms such as [[quicksort]].<ref>{{Citation|last1=Havil|first1=Julian|title=Gamma: Exploring Euler's Constant|publisher=[[Princeton University Press]]|isbn=978-0-691-09983-5|year=2003}}, sections 11.5 and 13.8</ref>
   
  +
===Transcendence of the logarithm===
More efficient is:
 
  +
The logarithm is an example of a [[transcendental function]] and from a theoretical point of view, the [[Gelfond–Schneider theorem]] asserts that logarithms usually take "difficult" values. The formal statement relies on the notion of [[algebraic number]]s, which includes all [[rational number]]s, but also numbers such as the [[square root of 2]] or
  +
:<math>\sqrt{-5+\sqrt[3]{3 / 13}}.</math>
  +
[[Complex number]]s that are not algebraic are called [[transcendental number|transcendental]];<ref>{{citation|title=Selected papers on number theory and algebraic geometry|volume=172|first1=Katsumi|last1=Nomizu|location=Providence, RI|publisher=AMS Bookstore|year=1996|isbn=978-0-8218-0445-2|page=21|url=http://books.google.com/books?id=uDDxdu0lrWAC&pg=PA21}}</ref> for example, π and ''e'' are such numbers. [[Almost all]] complex numbers are transcendental. Using these notions, the Gelfond&ndash;Scheider theorem states that given two algebraic numbers ''a'' and ''b'', log<sub>''b''</sub>(''a'') is either a transcendental number or a rational number ''p'' / ''q'' (in which case ''a''<sup>''q''</sup> = ''b''<sup>''p''</sup>, so ''a'' and ''b'' were closely related to begin with).<ref>{{Citation|last1=Baker|first1=Alan|author1-link=Alan Baker (mathematician)|title=Transcendental number theory|publisher=[[Cambridge University Press]]|isbn=978-0-521-20461-3|year=1975}}, p. 10</ref>
   
  +
==Calculation==
:<math>\ln (z) = 2 \sum_{n=0}^\infty \frac{1}{2n+1} {\left ( \frac{z-1}{z+1} \right ) }^{2n+1}</math>
 
   
  +
Logarithms are easy to compute in some cases, such as {{nowrap begin}}log<sub>10</sub>(1,000) = 3{{nowrap end}}. In general, logarithms can be calculated using [[power series]] or the [[arithmetic-geometric mean]], or be retrieved from a precalculated [[logarithm table]] that provides a fixed precision.<ref>{{Citation | last1=Muller | first1=Jean-Michel | title=Elementary functions | publisher=Birkhäuser Boston | location=Boston, MA | edition=2nd | isbn=978-0-8176-4372-0 | year=2006}}, sections 4.2.2 (p. 72) and 5.5.2 (p. 95)</ref><ref>{{Citation|author=Hart, Cheney, Lawson et al.|year=1968|publisher=John Wiley|location=New York|title=Computer Approximations|series=SIAM Series in Applied Mathematics}}, section 6.3, p. 105&ndash;111</ref>
for positive ''z''. For example,
 
  +
[[Newton's method]], an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.<ref>{{Citation|last1=Zhang|first1=M.|last2=Delgado-Frias|first2=J.G.|last3=Vassiliadis|first3=S.|title=Table driven Newton scheme for high precision logarithm generation|url=http://ce.et.tudelft.nl/publicationfiles/363_195_00326783.pdf|doi=10.1049/ip-cdt:19941268 |journal=IEE Proceedings Computers & Digital Techniques|issn=1350-387|volume=141|year=1994|issue=5|pages=281–292}}, section 1 for an overview</ref> Using look-up tables, [[CORDIC]]-like methods can be used to compute logarithms if the only available operations are addition and [[Arithmetic shift|bit shifts]].<ref>{{Citation |url= |first=J. E.|last=Meggitt|title=Pseudo Division and Pseudo Multiplication Processes|journal=IBM Journal|month=April|year=1962|doi=10.1147/rd.62.0210}}</ref><ref>{{Citation |last=Kahan |first=W. |authorlink= William Kahan |title=Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials |date= May 20, 2001 |publisher= |journal= |doi= }}</ref> Moreover, the [[Binary logarithm#Algorithm|binary logarithm algorithm]] calculates lb(''x'') [[recursion|recursively]] based on repeated squarings of ''x'', taking advantage of the relation
  +
:<math>\log_2(x^2) = 2 \log_2 (x). \,</math>
   
  +
===Power series===
:<math>\frac{\frac{11}{9} - 1}{\frac{11}{9} + 1} = \frac{1}{10}.</math>
 
  +
;Taylor series
   
  +
[[File:Taylor approximation of natural logarithm.gif|right|thumb|The Taylor series of&nbsp;ln(''z'') centered at&nbsp;''z''&nbsp;=&nbsp;1. The animation shows the first&nbsp;10 approximations along with the 99th and 100th. The approximations will not converge beyond a distance of 1 from the center.|alt=An animation showing increasingly good approximations of the logarithm graph.]]
So we get
 
  +
For any real number ''z'' that satisfies {{nowrap|0 < ''z'' < 2}}, the following formula holds:{{#tag:ref|The same series holds for the principal value of the complex logarithm for complex numbers ''z'' satisfying <nowiki>|</nowiki>''z'' &minus; 1<nowiki>|</nowiki> < 1.|group=nb}}<ref name=AbramowitzStegunp.68>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 68}}</ref>
  +
:<math>
  +
\ln (z) = (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots
  +
</math>
  +
This is a shorthand for saying that ln(''z'') can be approximated to a more and more accurate value by the following expressions:
  +
:<math>
  +
\begin{array}{lllll}
  +
(z-1) & & \\
  +
(z-1) & - & \frac{(z-1)^2}{2} & \\
  +
(z-1) & - & \frac{(z-1)^2}{2} & + & \frac{(z-1)^3}{3} \\
  +
\vdots &
  +
\end{array}
  +
</math>
  +
For example, with {{nowrap|''z'' {{=}} 1.5}} the third approximation yields 0.4167, which is about 0.011 greater than {{nowrap|ln(1.5) {{=}} 0.405465}}. This [[series (mathematics)|series]] approximates ln(''z'') with arbitrary precision, provided the number of summands is large enough. In elementary calculus, ln(''z'') is therefore the [[limit (mathematics)|''limit'']] of this series. It is the [[Taylor series]] of the natural logarithm at {{nowrap begin}}''z'' = 1{{nowrap end}}. The Taylor series of ln ''z'' provides a particularly useful approximation to ln(1+''z'') when ''z'' is small, ''|z| << 1'', since then
  +
:<math>
  +
\ln (1+z) = z - \frac{z^2}{2} + \cdots \approx z.
  +
</math>
  +
For example, with ''z'' = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.
   
  +
;More efficient series
:<math>\ln (1.2222222...) = \frac{1}{5} \left (1 + \frac{1}{3\cdot 100} + \frac{1}{5 \cdot 10000} +
 
  +
Another series is based on the [[area hyperbolic tangent]] function:
\frac{1}{7 \cdot 1000000} + ... \right ) </math>
 
  +
:<math>
:<math>= 0.2 \cdot (1.0000000... + 0.0033333... + 0.0000200... + 0.0000001... + ...)</math>
 
  +
\ln (z) = 2\cdot\operatorname{artanh}\,\frac{z-1}{z+1} = 2 \left ( \frac{z-1}{z+1} + \frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3 + \frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5 + \cdots \right ),
:<math>= 0.2 \cdot 1.0033535... = 0.2006707... </math>
 
  +
</math>
where we factored 1/10 out of the sum in the first line.
 
  +
for any real number ''z'' > 0.{{#tag:ref|The same series holds for the principal value of the complex logarithm for complex numbers ''z'' with positive real part.|group=nb}}<ref name=AbramowitzStegunp.68 /> Using the [[Summation#Capital-sigma notation|Sigma notation]], this is also written as
  +
:<math>\ln (z) = 2\sum_{n=0}^\infty\frac{1}{2n+1}\left(\frac{z-1}{z+1}\right)^{2n+1}.</math>
  +
This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if ''z'' is close to 1. For example, for {{nowrap begin}}''z'' = 1.5{{nowrap end}}, the first three terms of the second series approximate ln(1.5) with an error of about {{val|3|e=-6}}. The quick convergence for ''z'' close to 1 can be taken advantage of in the following way: given a low-accuracy approximation {{nowrap|''y'' &asymp; ln(''z'')}} and putting
  +
:<math>A = \frac z{\exp(y)}, \,</math>
  +
the logarithm of ''z'' is:
  +
:<math>\ln (z)=y+\ln (A). \,</math>
  +
The better the initial approximation ''y'' is, the closer ''A'' is to 1, so its logarithm can be calculated efficiently. ''A'' can be calculated using the [[exponential function|exponential series]], which converges quickly provided ''y'' is not too large. Calculating the logarithm of larger ''z'' can be reduced to smaller values of ''z'' by writing {{nowrap|''z'' {{=}} ''a'' · 10<sup>''b''</sup>}}, so that {{nowrap|ln(''z'') {{=}} ln(''a'') + ''b'' · ln(10)}}.
   
  +
A closely related method can be used to compute the logarithm of integers. From the above series, it follows that:
For any other base ''b'', we use
 
  +
:<math>\ln (n+1) = \ln(n) + 2\sum_{k=0}^\infty\frac{1}{2k+1}\left(\frac{1}{2 n+1}\right)^{2k+1}.</math>
  +
If the logarithm of a large integer ''n'' is known, then this series yields a fast converging series for log(''n''+1).
   
  +
===Arithmetic-geometric mean approximation===
:<math>\log_b (x) = \frac{\ln (x)}{\ln (b)}.</math>
 
  +
The [[arithmetic-geometric mean]] yields high precision approximations of the natural logarithm. ln(''x'') is approximated to a precision of 2<sup>&minus;''p''</sup> (or ''p'' precise bits) by the following formula (due to [[Carl Friedrich Gauss]]):<ref>{{Citation |first1=T. |last1=Sasaki |first2=Y. |last2=Kanada |title=Practically fast multiple-precision evaluation of log(x) |journal=Journal of Information Processing |volume=5|issue=4 |pages=247–250 |year=1982 | url=http://ci.nii.ac.jp/naid/110002673332 | accessdate=30 March 2011}}</ref><ref>{{Citation |first1=Timm |last1=Ahrendt|title=Fast computations of the exponential function|publisher=Springer|location=Berlin, New York|series=Lecture notes in computer science|doi=10.1007/3-540-49116-3_28|volume=1564|year=1999|pages=302–312}}</ref>
   
  +
:<math>\ln (x) \approx \frac{\pi}{2 M(1,2^{2-m}/x)} - m \ln (2).</math>
== Generalizations ==
 
Logarithms may also be defined for [[complex number|complex]] arguments. The logarithm (to base ''e'') of a complex number ''z'' is the complex number ln(|''z''|) + ''i'' arg(''z''), where |''z''| is the [[Absolute value#Complex numbers|modulus]] of ''z'', arg(''z'') is the [[Complex number#The complex plane|argument]], and ''i'' is the [[imaginary unit]]; see [[complex logarithm]] for details.
 
   
  +
Here ''M'' denotes the arithmetic-geometric mean. It is obtained by repeatedly calculating the average ([[arithmetic mean]]) and the square root of the product of two numbers ([[geometric mean]]). Moreover, ''m'' is chosen such that
The [[discrete logarithm]] is a related notion in the theory of [[finite group]]s. It involves solving the equation ''b''<sup>''n''</sup> = ''x'', where ''b'' and ''x'' are elements of the group, and ''n'' is an integer specifying a power in the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in [[public key cryptography]].
 
   
  +
:<math>x \,2^m > 2^{p/2}.\, </math>
The [[logarithm of a matrix]] is the inverse of the [[matrix exponential]].
 
   
  +
Both the arithmetic-geometric mean and the constants π and ln(2) can be calculated with quickly converging series.
A '''double logarithm''' is the inverse function of the [[Exponential function#Double exponential function|double-exponential function]]. A '''super-logarithm''' or '''hyper-logarithm''' is the inverse function of the [[Tetration#Extension to real numbers|super-exponential function]]. The super-logarithm of ''x'' grows even more slowly than the double logarithm for large ''x''.
 
   
  +
==Applications==
For each positive ''b'' not equal to 1, the function log<sub>''b'' </sub> (x) is an [[isomorphism]] from the [[group (mathematics)|group]] of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms. The logarithm function can be extended to a [[Haar measure]] in the [[topological group]] of positive real numbers under multiplication.
 
  +
[[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|right|A nautilus displaying a logarithmic spiral|alt=A photograph of a nautilus' shell.]]
  +
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of [[scale invariance]]. For example, each chamber of the shell of a [[nautilus]] is an approximate copy of the next one, scaled by a constant factor. This gives rise to a [[logarithmic spiral]].<ref>{{Harvard citations
  +
|last1=Maor
  +
|year=2009
  +
|nb=yes
  +
|loc=p. 135
  +
}}</ref> [[Benford's law]] on the distribution of leading digits can also be explained by scale invariance.<ref>{{Citation | last1=Frey | first1=Bruce | title=Statistics hacks | publisher=[[O'Reilly Media|O'Reilly]]|location=Sebastopol, CA| series=Hacks Series |url=http://books.google.com/?id=HOPyiNb9UqwC&pg=PA275&dq=statistics+hacks+benfords+law#v=onepage&q&f=false| isbn=978-0-596-10164-0 | year=2006}}, chapter 6, section 64</ref> Logarithms are also linked to [[self-similarity]]. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.<ref>{{Citation | last1=Ricciardi | first1=Luigi M. | title=Lectures in applied mathematics and informatics | url=http://books.google.de/books?id=Cw4NAQAAIAAJ | publisher=Manchester University Press | location=Manchester | isbn=978-0-7190-2671-3 | year=1990}}, p. 21, section 1.3.2</ref> The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.
  +
[[Logarithmic scale]]s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(''x'') grows very slowly for large ''x'', logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the [[Tsiolkovsky rocket equation]], the [[Fenske equation]], or the [[Nernst equation]].
   
  +
===Logarithmic scale===
== History ==
 
  +
{{Main|Logarithmic scale}}
In the [[17th century]], [[Joost Bürgi]], a [[Swiss]] clockmaker in the employ of the Duke of Hesse-Kassel, first discovered logarithms as a computational tool; however he did not publish his discovery until [[1620]]. The method of logarithms was first publicly propounded in [[1614]], in a book entitled ''Mirifici Logarithmorum Canonis Descriptio,'' by [[John Napier]], Baron of Merchiston in [[Scotland]], four years after the publication of his memorable discovery. This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved ''[[prosthaphaeresis]]'', which relied on [[trigonometric identities]], as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in the higher theoretical mathematics.
 
  +
[[File:GermanyHyperChart.jpg|A logarithmic chart depicting the value of one [[German gold mark|Goldmark]] in [[German Papiermark|Papiermarks]] during the [[Inflation in the Weimar Republic|German hyperinflation in the 1920s]]|right|thumb|alt=A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.]]
  +
Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, the [[decibel]] is a logarithmic unit of measurement. It is based on the common logarithm of [[ratio]]s&mdash;10 times the common logarithm of a [[power (physics)|power]] ratio or 20 times the common logarithm of a [[voltage]] ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals,<ref>{{Citation|last1=Bakshi|first1=U. A.|title=Telecommunication Engineering |publisher=Technical Publications|location=Pune|isbn=978-81-8431-725-1|year=2009|url=http://books.google.com/books?id=EV4AF0XJO9wC&pg=SA5-PA1#v=onepage&f=false}}, section 5.2</ref> to describe power levels of sounds in [[acoustics]],<ref>{{Citation|last1=Maling|first1=George C.|editor1-last=Rossing|editor1-first=Thomas D.|title=Springer handbook of acoustics|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-0-387-30446-5|year=2007|chapter=Noise}}, section 23.0.2</ref> and the [[absorbance]] of light in the fields of [[spectrometer|spectrometry]] and [[optics]]. The [[signal-to-noise ratio]] describing the amount of unwanted [[noise (electronic)|noise]] in relation to a (meaningful) [[signal (information theory)|signal]] is also measured in decibels.<ref>{{Citation | last1=Tashev | first1=Ivan Jelev | title=Sound Capture and Processing: Practical Approaches | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-470-31983-3 | year=2009|url=http://books.google.com/books?id=plll9smnbOIC&pg=PA48#v=onepage&f=false}}, p. 48</ref> In a similar vein, the [[peak signal-to-noise ratio]] is commonly used to assess the quality of sound and [[image compression]] methods using the logarithm.<ref>{{Citation | last1=Chui | first1=C.K. | title=Wavelets: a mathematical tool for signal processing | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | series=SIAM monographs on mathematical modeling and computation | isbn=978-0-89871-384-8 | year=1997|url=http://books.google.com/books?id=N06Gu433PawC&pg=PA180#v=onepage&f=false}}, p. 180</ref>
   
  +
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the [[moment magnitude scale]] or the [[Richter scale]]. For example, a 5.0 earthquake releases 10 times and a 6.0 releases 100 times the energy of a 4.0.<ref>{{Citation|last1=Crauder|first1=Bruce|last2=Evans|first2=Benny|last3=Noell|first3=Alan|title=Functions and Change: A Modeling Approach to College Algebra|publisher=Cengage Learning|location=Boston|edition=4th|isbn=978-0-547-15669-9|year=2008}}, section 4.4.</ref> Another logarithmic scale is [[apparent magnitude]]. It measures the brightness of stars logarithmically.<ref>{{Citation|last1=Bradt|first1=Hale|title=Astronomy methods: a physical approach to astronomical observations|publisher=[[Cambridge University Press]]|series=Cambridge Planetary Science|isbn=978-0-521-53551-9|year=2004}}, section 8.3, p. 231</ref> Yet another example is [[pH]] in [[chemistry]]; pH is the negative of the common logarithm of the [[Activity (chemistry)|activity]] of [[hydronium]] ions (the form [[hydrogen]] [[ion]]s {{chem|H|+|}} take in water).<ref>{{Citation|author=[[IUPAC]]|title=Compendium of Chemical Terminology ("Gold Book")|edition=2nd|editor=A. D. McNaught, A. Wilkinson|publisher=Blackwell Scientific Publications|location=Oxford|year=1997|url=http://goldbook.iupac.org/P04524.html|isbn=978-0-9678550-9-7|doi=10.1351/goldbook}}</ref> The activity of hydronium ions in neutral water is 10<sup>−7</sup>&nbsp;[[molar concentration|mol·L<sup>−1</sup>]], hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10<sup>4</sup> of the activity, that is, vinegar's hydronium ion activity is about 10<sup>−3</sup>&nbsp;mol·L<sup>−1</sup>.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word ''logarithm'' to mean a number that indicates a ratio: {{polytonic|λόγος}} (''[[logos]]'') meaning proportion, and {{polytonic|ἀριθμός}} (''arithmos'') meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an [[arithmetic series]] of logarithms corresponds to a [[geometric series]] of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.
 
   
  +
[[Semi-log plot|Semilog]] (log-linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, [[exponential function]]s of the form {{nowrap begin}}''f''(''x'') = ''a'' · ''b''<sup>''x''</sup>{{nowrap end}} appear as straight lines with [[slope]] equal to the logarithm of ''b''.
Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base <math>1/e</math>. For interpolation purposes and ease of calculation, it is useful to make the ratio <math>r</math> in the geometric series close to 1. Napier chose <math>r=1-10^{-7}=0.999999</math>, and Bürgi chose <math>r=1+10^{-4}=1.0001</math>. Napier's original logarithms did not have log 1 = 0 but rather log <math>10^7</math> = 0. Thus if <math>N</math> is a number and <math>L</math> is its logarithm as calculated by Napier, <math>N=10^7(1-10^{-7})^L</math>. Since <math>(1-10^{-7})^{10^7}</math> is approximately <math>1/e</math>, this makes <math>L/10^7</math> approximately equal to <math>\log_{(1/e)} N/10^7</math>. [http://mathforum.org/library/drmath/view/52469.html]
 
  +
[[Log-log plot|Log-log]] graphs scale both axes logarithmically, which causes functions of the form {{nowrap begin}}''f''(''x'') = ''a'' · ''x''<sup>''k''</sup>{{nowrap end}} to be depicted as straight lines with slope equal to the exponent ''k''. This is applied in visualizing and analyzing [[power law]]s.<ref>{{Citation|last1=Bird|first1=J. O.|title=Newnes engineering mathematics pocket book |publisher=Newnes|location=Oxford|edition=3rd|isbn=978-0-7506-4992-6|year=2001}}, section 34</ref>
   
=== Tables of logarithms ===
+
===Psychology===
  +
Logarithms occur in several laws describing [[human perception]]:<ref>{{Citation | last1=Goldstein | first1=E. Bruce | title=Encyclopedia of Perception | url=http://books.google.de/books?id=Y4TOEN4f5ZMC | publisher=Sage | location=Thousand Oaks, CA | series=Encyclopedia of Perception | isbn=978-1-4129-4081-8 | year=2009}}, p. 355&ndash;356</ref><ref>{{Citation | last1=Matthews | first1=Gerald | title=Human performance: cognition, stress, and individual differences | url=http://books.google.de/books?id=0XrpulSM1HUC | publisher=Psychology Press | location=Hove | series=Human Performance: Cognition, Stress, and Individual Differences | isbn=978-0-415-04406-6 | year=2000}}, p. 48</ref>
[[Image:Abramowitz&Stegun.page97.agr.jpg|thumb|Part of a 20th century table of [[common logarithm]]s in the reference book [[Abramowitz and Stegun]].]]
 
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[[Hick's law]] proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have.<ref>{{Citation|last1=Welford|first1=A. T.|title=Fundamentals of skill|publisher=Methuen|location=London|isbn=978-0-416-03000-6 |oclc=219156|year=1968}}, p. 61</ref> [[Fitts's law]] predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target.<ref>{{Citation|author=Paul M. Fitts|year=1954|title=The information capacity of the human motor system in controlling the amplitude of movement|journal=Journal of Experimental Psychology|volume=47|issue=6|month=June|pages=381–391 | pmid=13174710 | doi =10.1037/h0055392 }}, reprinted in {{Citation|journal=Journal of Experimental Psychology: General|volume=121|issue=3|pages=262&ndash;269|year=1992 | pmid=1402698 | url=http://sing.stanford.edu/cs303-sp10/papers/1954-Fitts.pdf | format=PDF | accessdate=30 March 2011 |title=The information capacity of the human motor system in controlling the amplitude of movement|author=Paul M. Fitts|doi=10.1037/0096-3445.121.3.262}}</ref> In [[psychophysics]], the [[Weber–Fechner law]] proposes a logarithmic relationship between [[stimulus (psychology)|stimulus]] and [[sensation (psychology)|sensation]] such as the actual vs. the perceived weight of an item a person is carrying.<ref>{{Citation | last1=Banerjee | first1=J. C. | title=Encyclopaedic dictionary of psychological terms | publisher=M.D. Publications | location=New Delhi | isbn=978-81-85880-28-0 | oclc=33860167 | year=1994|url=http://books.google.com/?id=Pwl5U2q5hfcC&pg=PA306&dq=weber+fechner+law#v=onepage&q=weber%20fechner%20law&f=false}}, p. 304</ref> (This "law", however, is less precise than more recent models, such as the [[Stevens' power law]].<ref>{{Citation|last1=Nadel|first1=Lynn|author1-link=Lynn Nadel|title=Encyclopedia of cognitive science|publisher=[[John Wiley & Sons]]|location=New York|isbn=978-0-470-01619-0|year=2005}}, lemmas ''Psychophysics'' and ''Perception: Overview''</ref>)
Prior to the advent of [[computer]]s and [[calculator]]s, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See [[common logarithm]] for details, including the use of characteristics and [[mantissa]]s of common (i.e., base-10) logarithms.
 
   
  +
Psychological studies found that mathematically unsophisticated individuals tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 20 as 100 is to 200. Increasing mathematical understanding shifts this to a linear estimate (positioning 100 10x as far away).<ref>{{Citation | doi=10.1111/1467-9280.02438 | last1=Siegler|first1=Robert S.|last2=Opfer|first2=John E.|title=The Development of Numerical Estimation. Evidence for Multiple Representations of Numerical Quantity|volume=14|issue=3|pages=237–43|year=2003|journal=Psychological Science
In [[1617]], [[Henry Briggs (mathematician)|Henry Briggs]] published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight [[decimal]] places. This he followed, in [[1624]], by his ''Arithmetica Logarithmica,'' containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by [[Adriaan Vlacq]], a [[the Netherlands|Dutch]] computer; but in his table, which appeared in [[1628]], the logarithms were given to only ten places of decimals.
 
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|url=http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf | pmid=12741747}}</ref><ref>{{Citation|last1=Dehaene| first1=Stanislas|last2=Izard|first2=Véronique |last3=Spelke| first3=Elizabeth|last4=Pica| first4=Pierre| title=Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures|volume=320|issue=5880|pages=1217–1220|doi=10.1126/science.1156540|pmc=2610411|pmid=18511690| year=2008|journal=Science|postscript=<!--None-->}}</ref>
   
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===Probability theory and statistics===
Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error." (''Athenaeum,'' [[15 June]] [[1872]]. See also the ''Monthly Notices of the Royal Astronomical Society'' for May [[1872]].) An edition of Vlacq's work, containing many corrections, was issued at [[Leipzig]] in [[1794]] under the title ''Thesaurus Logarithmorum Completus'' by [[Jurij Vega]].
 
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[[File:Some log-normal distributions.svg|thumb|right|alt=Three asymmetric PDF curves|Three [[probability density function]]s (PDF) of random variables with log-normal distributions. The location parameter {{math|&mu;}}, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.]]
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[[File:Benfords law illustrated by world's countries population.png|Distribution of first digits (in %, red bars) in the [[List of countries by population|population of the 237 countries]] of the world. Black dots indicate the distribution predicted by Benford's law.|thumb|right|alt=A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion.]]
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Logarithms arise in [[probability theory]]: the [[law of large numbers]] dictates that, for a [[fair coin]], as the number of coin-tosses increases to infinity, the observed proportion of heads [[binomial distribution#Symmetric binomial distribution (p = 0.5)|approaches one-half]]. The fluctuations of this proportion about one-half are described by the [[law of the iterated logarithm]].<ref>{{Citation | last1=Breiman | first1=Leo | title=Probability | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | series=Classics in applied mathematics | isbn=978-0-89871-296-4 | year=1992}}, section 12.9</ref>
   
  +
Logarithms also occur in [[log-normal distribution]]s. When the logarithm of a [[random variable]] has a [[normal distribution]], the variable is said to have a log-normal distribution.<ref>{{Citation|last1=Aitchison|first1=J.|last2=Brown|first2=J. A. C.|title=The lognormal distribution|publisher=[[Cambridge University Press]]|isbn=978-0-521-04011-2 |oclc=301100935|year=1969}}</ref> Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.<ref>
[[François Callet]]'s seven-place table ([[Paris]], [[1795]]), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of [[interpolation]], which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang [[1871]], whose table contained the seven-place logarithms of all numbers below 200,000.
 
  +
{{Citation
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| title = An introduction to turbulent flow
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| author = Jean Mathieu and Julian Scott
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| publisher = Cambridge University Press
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| year = 2000
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| isbn = 978-0-521-77538-0
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| page = 50
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| url = http://books.google.com/books?id=nVA53NEAx64C&pg=PA50
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}}</ref>
   
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Logarithms are used for [[maximum-likelihood estimation]] of parametric [[statistical model]]s. For such a model, the [[likelihood function]] depends on at least one [[parametric model|parameter]] that needs to be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "''log&nbsp;likelihood''"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for [[independence (probability)|independent]] random variables.<ref>{{Citation|last1=Rose|first1=Colin|last2=Smith|first2=Murray D.|title=Mathematical statistics with Mathematica|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Springer texts in statistics|isbn=978-0-387-95234-5|year=2002}}, section 11.3</ref>
Briggs and Vlacq also published original tables of the logarithms of the [[trigonometric function]]s.
 
   
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[[Benford's law]] describes the occurrence of digits in many [[data set]]s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is ''d'' (from 1 to 9) equals log<sub>10</sub>(''d'' + 1) − log<sub>10</sub>(''d''), ''regardless'' of the unit of measurement.<ref>{{Citation|last1=Tabachnikov|first1=Serge|title=Geometry and Billiards|publisher=[[American Mathematical Society]]|location=Providence, R.I.|isbn=978-0-8218-3919-5|year=2005|pages=36–40}}, section 2.1</ref> Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.<ref>{{Citation|title=The Effective Use of Benford's Law in Detecting Fraud in Accounting Data|first1=Cindy|last1=Durtschi| first2=William|last2=Hillison|first3=Carl|last3=Pacini|url=http://www.auditnet.org/articles/JFA-V-1-17-34.pdf| volume=V |pages=17–34|year=2004|journal=Journal of Forensic Accounting}}</ref>
Besides the tables mentioned above, a great collection, called ''Tables du Cadastre,'' was constructed under the direction of [[Gaspard de Prony]], by an original computation, under the auspices of the [[France|French]] republican government of the [[1700s]]. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in [[1792]]; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." (''English Cyclopaedia, Biography,'' Vol. IV., article "Prony.") [[Cubic function|Cubic]] [[interpolation]] could be used to find the logarithm of any number to a similar accuracy.
 
   
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===Computational complexity===
To the modern student who has the benefit of a calculator, the work put into the tables just mentioned is a small indication of the importance of logarithms.
 
   
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[[Analysis of algorithms]] is a branch of [[computer science]] that studies the [[time complexity|performance]] of [[algorithm]]s (computer programs solving a certain problem).<ref name=Wegener>{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005}}, pages 1-2</ref> Logarithms are valuable for describing algorithms which [[Divide and conquer algorithm|divide a problem]] into smaller ones, and join the solutions of the subproblems.<ref>{{Citation|last1=Harel|first1=David|last2=Feldman|first2=Yishai A.|title=Algorithmics: the spirit of computing|location=New York|publisher=[[Addison-Wesley]]|isbn=978-0-321-11784-7|year=2004}}, p. 143</ref>
== Trivia ==
 
=== Unicode glyph ===
 
''log'' has its own [[Unicode]] [[glyph]]: ㏒ (U+33D2 or 13266 in [[decimal]]). This is more likely due to its presence in Asian [[legacy encoding]]s than its importance as a mathematical function.
 
   
  +
For example, to find a number in a sorted list, the [[binary search algorithm]] checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log<sub>2</sub>(''N'') comparisons, where ''N'' is the list's length.<ref>{{citation | last = Knuth | first = Donald | authorlink = Donald Knuth | title = [[The Art of Computer Programming]] | publisher = Addison-Wesley |location=Reading, Mass. | year= 1998| isbn = 978-0-201-89685-5 }}, section 6.2.1, pp. 409–426</ref> Similarly, the [[merge sort]] algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time [[big O notation|approximately proportional to]] {{nowrap|''N'' · log(''N'')}}.<ref>{{Harvard citations|last = Knuth | first = Donald|year=1998|loc=section 5.2.4, pp. 158–168|nb=yes}}</ref> The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor, is usually disregarded in the analysis of algorithms under the standard [[uniform cost model]].<ref name=Wegener20>{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005|page=20}}</ref>
===Graphical interpretation===
 
The natural logarithm of ''a'' is the area under the curve ''y'' = 1/''x'' between the ''x'' values 1 and ''a''.
 
   
  +
A function ''f''(''x'') is said to [[Logarithmic growth|grow logarithmically]] if ''f''(''x'') is (exactly or approximately) proportional to the logarithm of ''x''. (Biological descriptions of organism growth, however, use this term for an exponential function.<ref>{{Citation |last1=Mohr|first1=Hans|last2=Schopfer|first2=Peter|title=Plant physiology|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-58016-4|year=1995}}, chapter 19, p. 298</ref>) For example, any [[natural number]] ''N'' can be represented in [[Binary numeral system|binary form]] in no more than {{nowrap|log<sub>2</sub>(''N'') + 1}} [[bit]]s. In other words, the amount of [[memory (computing)|memory]] needed to store ''N'' grows logarithmically with ''N''.
===Irrationality===
 
For [[integer]]s ''b'' and ''x'', the number log<sub>''b''</sub>(''x'') is [[irrational number|irrational]] (i.e., not a quotient of two integers) if one of ''b'' and ''x'' has a [[prime factor]] which the other does not. In certain cases this fact can be proved very quickly: for example, if log<sub>2</sub>3 were rational, we would have log<sub>2</sub>3 = ''n''/''m'' for some positive integers ''n'' and ''m'', thus implying 2<sup>''n''</sup> = 3<sup>''m''</sup>. But this last identity is impossible, since 2<sup>''n''</sup> is even and 3<sup>''m''</sup> is odd. Much stronger results are known. See [[Lindemann–Weierstrass theorem]].
 
   
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===Entropy and chaos===
=== Relationships between binary, natural and common logarithms ===
 
  +
[[File:Chaotic Bunimovich stadium.png|right|thumb|[[Dynamical billiards|Billiards]] on an oval [[billiard table]]. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of [[reflection (physics)|reflection]]s at the boundary.|alt=An oval shape with the trajectories of two particles.]]
In particular we have:
 
  +
:{|
 
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[[Entropy]] is broadly a measure of the disorder of some system. In [[statistical thermodynamics]], the entropy ''S'' of some physical system is defined as
| log<sub>2</sub>(''e'') ||≈ 1.44269504
 
  +
:<math> S = - k \sum_i p_i \ln(p_i).\, </math>
|-
 
  +
The sum is over all possible states ''i'' of the system in question, such as the positions of gas particles in a container. Moreover, ''p''<sub>''i''</sub> is the probability that the state ''i'' is attained and ''k'' is the [[Boltzmann constant]]. Similarly, [[entropy (information theory)|entropy in information theory]] measures the quantity of information. If a message recipient may expect any one of ''N'' possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log<sub>2</sub>(''N'') bits.<ref>{{Citation|last1=Eco|first1=Umberto|author1-link=Umberto Eco|title=The open work |publisher=[[Harvard University Press]]|isbn=978-0-674-63976-8|year=1989}}, section III.I</ref>
| log<sub>2</sub>(10) ||≈ 3.32192809
 
  +
  +
[[Lyapunov exponent]]s use logarithms to gauge the degree of chaoticity of a [[dynamical system]]. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are [[chaos theory|chaotic]] in a [[Deterministic system|deterministic]] way because small errors of measurement of the initial state will predictably lead to largely different final states.<ref>{{Citation | last1=Sprott | first1=Julien Clinton | title=Elegant Chaos: Algebraically Simple Chaotic Flows | url=http://books.google.com/books?id=buILBDre9S4C | publisher=[[World Scientific]] |location=New Jersey|isbn=978-981-283-881-0| year=2010}}, section 1.9</ref> At least one Lyapunov exponent of a deterministically chaotic system is positive.
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===Fractals===
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  +
[[File:Sierpinski dimension.svg|The Sierpinski triangle (at the right) is constructed by repeatedly replacing [[equilateral triangle]]s by three smaller ones.|right|thumb|400px|alt=Parts of a triangle are removed in an iterated way.]]
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Logarithms occur in definitions of the [[fractal dimension|dimension]] of [[fractal]]s.<ref>{{Citation|last1=Helmberg|first1=Gilbert|title=Getting acquainted with fractals|publisher=Walter de Gruyter|series=De Gruyter Textbook|location=Berlin, New York|isbn=978-3-11-019092-2|year=2007}}</ref> Fractals are geometric objects that are [[self-similarity|self-similar]]: small parts reproduce, at least roughly, the entire global structure. The [[Sierpinski triangle]] (pictured) can be covered by three copies of itself, each having sides half the original length. This causes the [[Hausdorff dimension]] of this structure to be {{nowrap begin}}log(3)/log(2) ≈ 1.58{{nowrap end}}. Another logarithm-based notion of dimension is obtained by [[box-counting dimension|counting the number of boxes]] needed to cover the fractal in question.
  +
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===Music===
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Logarithms are related to musical tones and [[interval (music)|intervals]]. In [[equal temperament]], the frequency ratio depends only on the interval between two tones, not on the specific frequency, or [[pitch (music)|pitch]], of the individual tones. For example, the [[a (musical note)|note ''A'']] has a frequency of 440 [[Hertz|Hz]] and [[B♭ (musical note)|''B-flat'']] has a frequency of 466&nbsp;Hz. The interval between ''A'' and ''B-flat'' is a [[semitone]], as is the one between ''B-flat'' and [[b (musical note)|''B'']] (frequency 493&nbsp;Hz). Accordingly, the frequency ratios agree:
  +
:<math>\frac{466}{440} \approx \frac{493}{466} \approx 1.059 \approx \sqrt[12]2.</math>
  +
Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the {{nowrap|base-2<sup>1/12</sup>}} logarithm of the [[frequency]] ratio, while the {{nowrap|base-2<sup>1/1200</sup>}} logarithm of the frequency ratio expresses the interval in [[cent (music)|cents]], hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.<ref>{{Citation|last1=Wright|first1=David|title=Mathematics and music|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4873-9|year=2009}}, chapter 5</ref>
  +
  +
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
 
|-
 
|-
  +
||'''Interval'''<br>(the two tones are played at the same time)
| log<sub>''e''</sub>(10) ||≈ 2.30258509
 
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||[[72 tone equal temperament|1/12 tone]] {{audio|1_step_in_72-et_on_C.mid|play}}
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||[[Semitone]] {{audio|help=no|Minor_second_on_C.mid|play}}
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||[[Just major third]] {{audio|help=no|Just_major_third_on_C.mid|play}}
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||[[Major third]] {{audio|help=no|Major_third_on_C.mid|play}}
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||[[Tritone]] {{audio|help=no|Tritone_on_C.mid|play}}
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||[[Octave]] {{audio|help=no|Perfect_octave_on_C.mid|play}}
 
|-
 
|-
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|| '''Frequency ratio''' ''r''
| log<sub>''e''</sub>(2) ||≈ 0.693147181
 
  +
|| <math>2^{\frac 1 {72}} \approx 1.0097</math>
  +
|| <math>2^{\frac 1 {12}} \approx 1.0595</math>
  +
|| <math>\tfrac 5 4 = 1.25</math>
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|| <math>\begin{align} 2^{\frac 4 {12}} & = \sqrt[3] 2 \\ & \approx 1.2599 \end{align} </math>
  +
|| <math>\begin{align} 2^{\frac 6 {12}} & = \sqrt 2 \\ & \approx 1.4142 \end{align} </math>
  +
|| <math> 2^{\frac {12} {12}} = 2 </math>
 
|-
 
|-
  +
|| '''Corresponding number of semitones'''<br><math>\log_{\sqrt[12] 2}(r) = 12 \log_2 (r)</math>
| log<sub>10</sub>(2) ||≈ 0.301029996
 
  +
|| <math>\tfrac 1 6 \,</math>
  +
|| <math>1 \,</math>
  +
|| <math>\approx 3.8631 \,</math>
  +
|| <math>4 \,</math>
  +
|| <math>6 \,</math>
  +
|| <math>12 \,</math>
 
|-
 
|-
  +
|| '''Corresponding number of cents'''<br><math>\log_{\sqrt[1200] 2}(r) = 1200 \log_2 (r)</math>
| log<sub>10</sub>(''e'') ||≈ 0.434294482
 
  +
|| <math>16 \tfrac 2 3 \,</math>
  +
|| <math>100 \,</math>
  +
|| <math>\approx 386.31 \,</math>
  +
|| <math>400 \,</math>
  +
|| <math>600 \,</math>
  +
|| <math>1200 \,</math>
 
|}
 
|}
A curious coincidence is the [[approximation]] log<sub>2</sub>(''x'') ≈ log<sub>10</sub>(''x'') + ln(''x''), accurate to about 99.4% or 2 [[significant digit]]s; this is because <sup>1</sup>/<sub>ln(2)</sub> &minus; <sup>1</sup>/<sub>ln(10)</sub> ≈ 1 (in fact 1.0084...).
 
   
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===Number theory===
Another interesting coincidence is that log<sub>10</sub>(2) ≈ 0.3 (the actual value is about 0.301029996); this corresponds to the fact that, with an error of only 2.4%, 2<sup>10</sup> ≈ 10<sup>3</sup>
 
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Natural logarithms are closely linked to [[prime-counting function|counting prime number]]s (2, 3, 5, 7, 11, ...), an important topic in [[number theory]]. For any [[integer]] ''x'', the quantity of [[prime number]]s less than or equal to ''x'' is denoted [[prime-counting function|π(''x'')]]. The [[prime number theorem]] asserts that π(''x'') is approximately given by
(i.e. 1024 is about 1000; see also [[binary prefix]]).
 
  +
:<math>\frac{x}{\ln(x)},</math>
  +
in the sense that the ratio of π(''x'') and that fraction approaches 1 when ''x'' tends to infinity.<ref>{{Citation|last1=Bateman|first1=P. T.|last2=Diamond|first2=Harold G.|title=Analytic number theory: an introductory course|publisher=[[World Scientific]]|location=New Jersey|isbn=978-981-256-080-3 |oclc=492669517|year=2004}}, theorem 4.1</ref> As a consequence, the probability that a randomly chosen number between 1 and ''x'' is prime is inversely [[proportionality (mathematics)|proportional]] to the numbers of decimal digits of ''x''. A far better estimate of π(''x'') is given by the
  +
[[logarithmic integral function|offset logarithmic integral]] function Li(''x''), defined by
  +
:<math> \mathrm{Li}(x) = \int_2^x \frac1{\ln(t)} \,dt. </math>
  +
The [[Riemann hypothesis]], one of the oldest open mathematical [[conjecture]]s, can be stated in terms of comparing π(''x'') and Li(''x'').<ref>{{Harvard citations|last1=Bateman|first1=P. T.|last2=Diamond|year=2004|nb=yes |loc=Theorem 8.15}}</ref> The [[Erdős–Kac theorem]] describing the number of distinct [[prime factor]]s also involves the natural logarithm.
   
  +
The logarithm of ''n'' [[factorial]], {{nowrap begin}}''n''! = 1 · 2 · ... · ''n''{{nowrap end}}, is given by
===Relation with positional notation===
 
  +
:<math> \ln (n!) = \ln (1) + \ln (2) + \cdots + \ln (n). \,</math>
[[Numeral system#Positional systems in detail | Positional notation]] is the way numbers are written with digits. The individual digits in the number are to be multiplied by a weight that going from right to left increases as 1,10,100,1000... for ''positions'' 0,1,2,3... in [[decimal|base 10]]. Positional notation contains a simple logarithmic function because the position is the common logarithm of the corresponding weight. This logaritmic function is however only sampled at integer spots.
 
  +
This can be used to obtain [[Stirling's formula]], an approximation of ''n''! for large ''n''.<ref>{{Citation|last1=Slomson|first1=Alan B.|title=An introduction to combinatorics|publisher=[[CRC Press]]|location=London|isbn=978-0-412-35370-3|year=1991}}, chapter 4</ref>
   
== See also ==
+
==Generalizations==
* [[List of logarithm topics]]
+
===Complex logarithm===
  +
{{Main|Complex logarithm}}
* [[Logarithmic identities]]
 
  +
[[File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of {{nowrap|''z {{=}} x + iy''}}. Both φ and φ' are arguments of ''z''.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the ''x'' axis.]]
* [[Logarithmic scale]]
 
* [[Natural logarithm]]
 
* [[Common logarithm]]
 
* [[Indefinite logarithm]]
 
* [[Logarithmic units]]
 
* [[Discrete logarithm]]
 
* [[Zech's logarithms]]
 
* [[Logarithm of a matrix]]
 
* [[Log-normal distribution]]
 
* [[Decibel]] (expressing loudness on a [[logarithmic scale]])
 
* [[Equal temperament]] classifying pitch on a [[logarithmic scale]]
 
* [[Richter scale]]
 
   
  +
The [[complex number]]s ''a'' solving the equation
== References ==
 
   
  +
:<math>e^a=z.\,</math>
http://www.giovannipastore.it/index_english.htm
 
   
  +
are called ''complex logarithms''. Here, ''z'' is a complex number. A complex number is commonly represented as {{nowrap begin}}''z = x + iy''{{nowrap end}}, where ''x'' and ''y'' are real numbers and ''i'' is the [[imaginary unit]]. Such a number can be visualized by a point in the [[complex plane]], as shown at the right. The [[polar form]] encodes a non-zero complex number ''z'' by its [[absolute value]], that is, the distance ''r'' to the [[origin (mathematics)|origin]], and an angle between the ''x'' axis and the line passing through the origin and ''z''. This angle is called the [[Argument (complex analysis)|argument]] of ''z''. The absolute value ''r'' of ''z'' is
Giovanni Pastore - Antikythera e i regoli calcolatori - Roma 2006 http://www.giovannipastore.it/ISTRUZIONI.htm
 
   
  +
:<math>r=\sqrt{x^2+y^2}. \,</math>
Much of the history of logarithms is derived from ''The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions'', by James Mills Peirce, University Professor of Mathematics in [[Harvard University]], [[1873]].
 
<references/>
 
   
  +
The argument is not uniquely specified by ''z'': both φ and φ' = φ + 2π are arguments of ''z'' because adding 2π [[radian]]s or 360 degrees{{#tag:ref|See [[radian]] for the conversion between 2[[pi|&pi;]] and 360 [[degree (angle)|degrees]].|group=nb}} to φ corresponds to "winding" around the origin counter-clock-wise by a [[Turn (geometry)|turn]]. The resulting complex number is again ''z'', as illustrated at the right. However, exactly one argument φ satisfies {{nowrap|−&pi; < &phi;}} and {{nowrap|&phi; &le; &pi;}}. It is called the ''principal argument'', denoted Arg(''z''), with a capital ''A''.<ref>{{Citation|last1=Ganguly|location=Kolkata|first1=S.|title=Elements of Complex Analysis|publisher=Academic Publishers|isbn=978-81-87504-86-3|year=2005}}, Definition 1.6.3</ref> (An alternative normalization is {{nowrap|0 &le; Arg(''z'') < 2&pi;}}.<ref>{{Citation|last1=Nevanlinna|first1=Rolf Herman|author1-link=Rolf Nevanlinna|last2=Paatero|first2=Veikko|title=Introduction to complex analysis|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4399-4|year=2007}}, section 5.9</ref>)
== External links ==
 
  +
* [http://www.mathlogarithms.com/ Explaining Logarithms]
 
  +
[[File:Complex log.jpg|right|thumb|The principal branch of the complex logarithm, Log(''z''). The black point at {{nowrap|''z'' {{=}} 1}} corresponds to absolute value zero and brighter (more [[saturation (color theory)|saturated]]) colors refer to bigger absolute values. The [[hue]] of the color encodes the argument of Log(''z'').|alt=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]
* [http://wolf.galekus.com/viewpage.php?page_id=10 Log Calculator for all bases.]
 
  +
Using [[trigonometric functions]] [[sine]] and [[cosine]], or the [[complex exponential]], respectively, ''r'' and φ are such that the following identities hold:<ref>{{Citation|last1=Moore|first1=Theral Orvis|last2=Hadlock|first2=Edwin H.|title=Complex analysis|publisher=[[World Scientific]]|location=Singapore|isbn=978-981-02-0246-0|year=1991}}, section 1.2</ref>
* [http://mathworld.wolfram.com/Logarithm.html Logarithm] on [[MathWorld]]
 
  +
* [http://www.micheloud.com/FXM/LOG/index.htm Jost Burgi, Swiss Inventor of Logarithms]
 
  +
:<math>\begin{array}{lll}z& = & r \left (\cos \varphi + i \sin \varphi\right) \\
* [http://www.algebra.com/algebra/homework/logarithm/ Logarithm calculators and word problems with work shown, for school students]
 
  +
& = & r e^{i \varphi}.
* [http://johnnapier.com/table_of_logarithms_001.htm Translation of Napier's work on logarithms]
 
  +
\end{array} \,
* [http://www.tufts.edu/~gdallal/logs.htm Logarithms - from The Little Handbook of Statistical Practice]
 
  +
</math>
* [http://en.literateprograms.org/Logarithm_Function_%28Python%29 Algorithm for determining Log values for any base]
 
  +
*[http://www.jacques-laporte.org/Briggs%20and%20the%20HP35.htm "Henry Briggs and the HP-35"], Jacques Laporte, Paris 2005
 
  +
This implies that the {{nowrap|''a''-th}} power of ''e'' equals ''z'', where
  +
  +
:<math>a = \ln (r) + i ( \varphi + 2 n \pi ), \,</math>
  +
  +
φ is the principal argument Arg(''z'') and ''n'' is an arbitrary integer. Any such ''a'' is called a complex logarithm of ''z''. There are infinitely many of them, in contrast to the uniquely defined real logarithm. If {{nowrap begin}}''n'' = 0{{nowrap end}}, ''a'' is called the ''principal value'' of the logarithm, denoted Log(''z''). The principal argument of any positive real number ''x'' is 0; hence Log(''x'') is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers [[Exponentiation#Failure of power and logarithm identities|do ''not'' generalize]] to the principal value of the complex logarithm.<ref>{{Citation | last1=Wilde | first1=Ivan Francis | title=Lecture notes on complex analysis | publisher=Imperial College Press | location=London | isbn=978-1-86094-642-4 | year=2006|url=http://books.google.com/?id=vrWES2W6vG0C&pg=PA97&dq=complex+logarithm#v=onepage&q=complex%20logarithm&f=false}}, theorem 6.1.</ref>
  +
  +
The illustration at the right depicts Log(''z''). The discontinuity, that is, the jump in the hue at the negative part of the ''x''- or real axis, is caused by the jump of the principal argument there. This locus is called a [[branch cut]]. This behavior can only be circumvented by dropping the range restriction on φ. Then the argument of ''z'' and, consequently, its logarithm become [[multi-valued function]]s.
  +
  +
===Inverses of other exponential functions===
  +
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the [[logarithm of a matrix]] is the (multi-valued) inverse function of the [[matrix exponential]].<ref>{{Citation|last1=Higham|first1=Nicholas|author1-link=Nicholas Higham|title=Functions of Matrices. Theory and Computation|location=Philadelphia, PA|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-646-7|year=2008}}, chapter 11.</ref> Another example is the [[p-adic logarithm function|''p''-adic logarithm]], the inverse function of the [[p-adic exponential function|''p''-adic exponential]]. Both are defined via Taylor series analogous to the real case.<ref>{{Neukirch ANT}}, section II.5.</ref> In the context of [[differential geometry]], the [[exponential map]] maps the [[tangent space]] at a point of a [[differentiable manifold|manifold]] to a [[neighborhood (mathematics)|neighborhood]] of that point. Its inverse is also called the logarithmic (or log) map.<ref>{{Citation|last1=Hancock|first1=Edwin R.|last2=Martin|first2=Ralph R.|last3=Sabin|first3=Malcolm A.|title=Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 Proceedings|url=http://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379|publisher=Springer|year=2009|page=379|isbn=978-3-642-03595-1}}</ref>
  +
  +
In the context of [[finite groups]] exponentiation is given by repeatedly multiplying one group element ''b'' with itself. The [[discrete logarithm]] is the integer ''n'' solving the equation
  +
:<math>b^n = x,\,</math>
  +
where ''x'' is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in [[public key cryptography]], such as for example in the [[Diffie–Hellman key exchange]], a routine that allows secure exchanges of [[cryptography|cryptographic]] keys over unsecured information channels.<ref>{{Citation|last1=Stinson|first1=Douglas Robert|title=Cryptography: Theory and Practice|publisher=[[CRC Press]]|location=London|edition=3rd|isbn=978-1-58488-508-5|year=2006}}</ref> [[Zech's logarithm]] is related to the discrete logarithm in the multiplicative group of non-zero elements of a [[finite field]].<ref>{{Citation|last1=Lidl|first1=Rudolf|last2=Niederreiter|first2=Harald|title=Finite fields|publisher=Cambridge University Press|isbn=978-0-521-39231-0|year=1997}}</ref>
  +
  +
Further logarithm-like inverse functions include the ''double logarithm'' ln(ln(''x'')), the ''[[super-logarithm|super- or hyper-4-logarithm]]'' (a slight variation of which is called [[iterated logarithm]] in computer science), the [[Lambert W function]], and the [[logit]]. They are the inverse functions of the [[double exponential function]], [[tetration]], of {{nowrap|''f''(''w'') {{=}} ''we<sup>w</sup>''}},<ref>{{Citation | last1=Corless | first1=R. | last2=Gonnet | first2=G. | last3=Hare | first3=D. | last4=Jeffrey | first4=D. | last5=Knuth | first5=Donald | author5-link=Donald Knuth | title=On the Lambert ''W'' function | url=http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1996 | journal=Advances in Computational Mathematics | issn=1019-7168 | volume=5 | pages=329–359 | doi=10.1007/BF02124750}}</ref> and of the [[logistic function]], respectively.<ref>{{Citation | last1=Cherkassky | first1=Vladimir | last2=Cherkassky | first2=Vladimir S. | last3=Mulier | first3=Filip | title=Learning from data: concepts, theory, and methods | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley series on adaptive and learning systems for signal processing, communications, and control | isbn=978-0-471-68182-3 | year=2007}}, p. 357</ref>
  +
  +
===Related concepts===
  +
From the perspective of [[pure mathematics]], the identity {{nowrap|log(''cd'') {{=}} log(''c'') + log(''d'')}} expresses a [[group isomorphism]] between positive [[real number|reals]] under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.<ref>{{Citation|last1=Bourbaki|first1=Nicolas|author1-link=Nicolas Bourbaki|title=General topology. Chapters 5—10|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Elements of Mathematics|isbn=978-3-540-64563-4|mr=1726872|year=1998}}, section V.4.1</ref> By means of that isomorphism, the [[Haar measure]] ([[Lebesgue measure]]) ''dx'' on the reals corresponds to the Haar measure ''dx''/''x'' on the positive reals.<ref>{{Citation|last1=Ambartzumian|first1=R. V.|title=Factorization calculus and geometric probability|publisher=[[Cambridge University Press]]|isbn=978-0-521-34535-4|year=1990}}, section 1.4</ref> In [[complex analysis]] and [[algebraic geometry]], [[differential form]]s of the form {{nowrap begin}}''df''/''f'' {{nowrap end}} are known as forms with logarithmic [[Pole (complex analysis)|pole]]s.<ref>{{Citation|last1=Esnault|first1=Hélène|last2=Viehweg|first2=Eckart|title=Lectures on vanishing theorems|location=Basel, Boston|publisher=Birkhäuser Verlag|series=DMV Seminar|isbn=978-3-7643-2822-1|mr=1193913|year=1992|volume=20}}, section 2</ref>
  +
  +
The [[polylogarithm]] is the function defined by
  +
:<math>
  +
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
  +
</math>
  +
It is related to the natural logarithm by {{nowrap begin}}Li<sub>1</sub>(''z'') = −ln(1 − ''z''){{nowrap end}}. Moreover, Li<sub>''s''</sub>(1) equals the [[Riemann zeta function]] ζ(''s'').<ref>{{dlmf|id= 25.12|first= T.M.|last= Apostol|ref= harv}}</ref>
  +
  +
==See also==
  +
* [[Index of logarithm articles]]
  +
* [[Exponential function]]
  +
  +
==Notes==
  +
{{reflist|group=nb|30em}}
  +
  +
==References==
  +
{{Reflist|30em}}
  +
  +
==External links==
  +
{{Commons category-inline}}
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* {{Citation|author=Colin Byfleet|url=http://mediasite.oddl.fsu.edu/mediasite/Viewer/?peid=003298f9a02f468c8351c50488d6c479|title=Educational video on logarithms|accessdate=12/10/2010}}
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* {{Citation|author=Edward Wright|url=http://johnnapier.com/table_of_logarithms_001.htm|title=Translation of Napier's work on logarithms|accessdate=12/10/2010}}
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{{Use dmy dates|date=February 2011}}
   
   
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[[Category:Scottish inventions]]
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File:Binary logarithm plot with ticks.svg

The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 × 10 × 10 = 103. More generally, if x = by , then y is the logarithm of x to base b, and is written y = logb(x), so log10(1000) = 3.


Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

The logarithm to base b = 10

is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant 
e (≈ 2.718

) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base b = 2

and is prominent in computer science.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. In chemistry, pH and pOH are logarithmic measures for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulae counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.

Motivation and definition

The idea of logarithms is to reverse the operation of exponentiation, that is raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2:

It follows that the logarithm of 8 with respect to base 2 is 3, so log2 8 = 3.

Exponentiation

The third power of some number b is the product of three factors of b. More generally, raising b to the n-th power, where n is a natural number, is done by multiplying n factors of b. The n-th power of b is written bn, so that

Exponentation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, 1/b.[nb 1]

Definition

The logarithm of a number x with respect to base b is the exponent by which b has to be raised to yield x. In other words, the logarithm of x to base b is the solution y to the equation[2]

The logarithm is denoted "logb(x)" (pronounced as "the logarithm of x to base b" or "the base-b logarithm of x"). In the equation y = logb(x), the value y is the answer to the question "To what power must b be raised, in order to yield x?". For the logarithm to be defined, the base b must be a positive real number not equal to 1 and x must be a positive number.[nb 2]

Examples

For example, log2(16) = 4, since 24 = 2 ×2 × 2 × 2 = 16. Logarithms can also be negative:

since

A third example: log10(150) is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000. Finally, for any base b, logb(b) = 1 and logb(1) = 0, since b1 = b and b0 = 1, respectively.

Logarithmic identities

Main article: List of logarithmic identities

Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another.[3]

Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. Therefore, the logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples:

Formula Example
product
quotient
power
root

Change of base

The logarithm logb(x) can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:

Typical scientific calculators calculate the logarithms to bases 10 and e.[4] Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

Given a number x and its logarithm logb(x) to an unknown base b, the base is given by:

Particular bases

Among all choices for the base b, three are particularly common. These are b = 10, b = e (the irrational mathematical constant ≈ 2.71828), and b = 2. In mathematical analysis, the logarithm to base e is widespread because of its particular analytical properties explained below. On the other hand, base-10 logarithms are easy to use for manual calculations in the decimal number system:[5]

Thus, log10(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log10(x).[6] For example, log10(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in computer science, where the binary system is ubiquitous.

The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(x) instead of logb(x), when the intended base can be determined from the context. The notation blog(x) also occurs.[7] The "ISO notation" column lists designations suggested by the International Organization for Standardization (ISO 31-11).[8]

Base b Name for logb(x) ISO notation Other notations Used in
2 binary logarithm lb(x)[9] ld(x), log(x), lg(x) computer science, information theory, mathematics
e natural logarithm ln(x)[nb 3] log(x)
(in mathematics and many programming languages[nb 4])
mathematical analysis, physics, chemistry,
statistics, economics, and some engineering fields
10 common logarithm lg(x) log(x)
(in engineering, biology, astronomy),
various engineering fields (see decibel and see below),
logarithm tables, handheld calculators

History

Predecessors

The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares.[13][14] However it could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards till superseded by the use of computers.

Michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains a table[15] of integers and powers of 2 that has been considered an early version of a logarithmic table.[16][17]

In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity

or similar to convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.

From Napier to Euler

File:John Napier.jpg

John Napier (1550–1617), the inventor of logarithms

The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).[18] Joost Bürgi independently invented logarithms but published six years after Napier.[19]

Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to John Napier,[20] remarked:

...the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ...instead of rearing up his child for the public benefit he deserted it in the birth.

Johannes Kepler[21], Rudolphine Tables (1627)

By repeated subtractions Napier calculated (1 − 10−7)L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10−5 . Napier then calculated the products of these numbers with 107(1 − 10−5)L for L from 1 to 50, and did similarly with 0.9998 ≈ (1 − 10−5)20 and 0.9 ≈ 0.99520. These computations, which occupied 20 years, allowed him to give, for any number N from 5 to 10 million, the number L that solves the equation

Napier first called L an "artificial number", but later introduced the word "logarithm" to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. In modern notation, the relation to natural logarithms is: [22]

where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes Kepler's Chilias logarithmorum (Germany) helped spread the concept further.[23]

File:1 over x integral.svg

The hyperbola y = 1/x (red curve) and the area from x = 1 to 6 (shaded in orange).

In 1647 Grégoire de Saint-Vincent related logarithms to the quadrature of the hyperbola, by pointing out that the area f(t) under the hyperbola from x = 1 to x = t satisfies

The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668,[24] although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm.[25] Around 1730, Leonhard Euler defined the exponential function and the natural logarithm by

Euler also showed that the two functions are inverse to one another.[26][27][28]

Logarithm tables, slide rules, and historical applications

File:Logarithms Britannica 1797.png

The 1797 Encyclopædia Britannica explanation of logarithms

By simplifying difficult calculations, logarithms contributed to the advance of science, and especially of astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms

[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.[29]

A key tool that enabled the practical use of logarithms before calculators and computers was the table of logarithms.[30] The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention. Subsequently, tables with increasing scope and precision were written. These tables listed the values of logb(x) and bx for any number x in a certain range, at a certain precision, for a certain base b (usually b = 10 ). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits. As the function f(x) = bx is the inverse function of logb(x), it has been called the antilogarithm.[31] The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c/d came from looking up the antilogarithm of the sum or difference, also via the same table:

and

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities. Calculations of powers and roots are reduced to multiplications or divisions and look-ups by

and

Many logarithm tables give logarithms by separately providing the characteristic and mantissa of x, that is to say, the integer part and the fractional part of log10(x).[32] The characteristic of 10 · x is one plus the characteristic of x, and their significands are the same. This extends the scope of logarithm tables: given a table listing log10(x) for all integers x ranging from 1 to 1000, the logarithm of 3542 is approximated by

Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation, as illustrated here:

File:Slide rule example2 with labels.svg

Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x.

The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.[26]

Analytic properties

A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number.[33] An example is the function producing the x-th power of b from any real number x, where the base (or radix) b is a fixed number. This function is written

Logarithmic function

To justify the definition of logarithms, it is necessary to show that the equation

has a solution x and that this solution is unique, provided that y is positive and that b is positive and unequal to 1. A proof of that fact requires the intermediate value theorem from elementary calculus.[34] This theorem states that a continuous function which produces two values m and n also produces any value that lies between m and n. A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen.

This property can be shown to hold for the function f(x) = bx . Because f takes arbitrarily large and arbitrarily small positive values, any number y > 0 lies between f(x0) and f(x1) for suitable x0 and x1. Hence, the intermediate value theorem ensures that the equation f(x) = y has a solution. Moreover, there is only one solution to this equation, because the function f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1).[35]

The unique solution x is the logarithm of y to base b, logb(y). The function which assigns to y its logarithm is called logarithm function or logarithmic function (or just logarithm).

Inverse function

File:Logarithm inversefunctiontoexp.svg

The graph of the logarithm function logb(x) (blue) is obtained by reflecting the graph of the function bx (red) at the diagonal line ( x = y ).

The formula for the logarithm of a power says in particular that for any number x,

In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positive number y, the formula

says that first taking the logarithm and then exponentiating gives back y. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base b is the inverse function of f(x) = bx.[36]

Inverse functions are closely related to the original functions. Their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown at the right: a point (t, u = bt) on the graph of f yields a point (u, t = logbu) on the graph of the logarithm and vice versa. As a consequence, logb(x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, logb(x) is an increasing function. For b < 1, logb(x) tends to minus infinity instead. When x approaches zero, logb(x) goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).

Derivative and antiderivative

File:Logarithm derivative.svg

The graph of the natural logarithm (green) and its tangent at x = 1.5 (black)

Analytic properties of functions pass to their inverses.[34] Thus, as f(x) = bx

is a continuous and differentiable function, so is logb(y). Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f(x) evaluates to ln(b)bx by the properties of the exponential function, the chain rule implies that the derivative of logb(x) is given by[35][37]

That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, logb(x)) equals 1/(x ln(b)). In particular, the derivative of ln(x) is 1/x, which implies that the antiderivative of 1/x is ln(x) + C. The derivative with a generalised functional argument f(x) is

The quotient at the right hand side is called the logarithmic derivative of f. Computing f'(x) by means of the derivative of ln(f(x)) is known as logarithmic differentiation.[38] The antiderivative of the natural logarithm ln(x) is:[39]

Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.[40]

Integral representation of the natural logarithm

File:Natural logarithm integral.svg

The natural logarithm of t is the shaded area underneath the graph of the function f(x) = 1/x (reciprocal of x).

The natural logarithm of t agrees with the integral of 1/x dx from 1 to t:

In other words, ln(t) equals the area between the x axis and the graph of the function 1/x, ranging from x = 1

to 

x = t

(figure at the right). This is a consequence of the fundamental theorem of calculus and the fact that derivative of ln(x) is 1/x. The right hand side of this equation can serve as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition.[41] For example, the product formula 

ln(tu) = ln(t) + ln(u)

is deduced as:

The equality (1) splits the integral into two parts, while the equality (2) is a change of variable ( w = x/t ). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function f(x) = 1/x

again. Therefore, the left hand blue area, which is the integral of f(x) from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.
File:Natural logarithm product formula proven geometrically.svg

A visual proof of the product formula of the natural logarithm

The power formula ln(tr) = r ln(t)

may be derived in a similar way:

The second equality uses a change of variables (integration by substitution), w := x1/r .

The sum over the reciprocals of natural numbers,

is called the harmonic series. It is closely tied to the natural logarithm: as n tends to infinity, the difference,

converges (i.e., gets arbitrarily close) to a number known as the Euler–Mascheroni constant. This relation aids in analyzing the performance of algorithms such as quicksort.[42]

Transcendence of the logarithm

The logarithm is an example of a transcendental function and from a theoretical point of view, the Gelfond–Schneider theorem asserts that logarithms usually take "difficult" values. The formal statement relies on the notion of algebraic numbers, which includes all rational numbers, but also numbers such as the square root of 2 or

Complex numbers that are not algebraic are called transcendental;[43] for example, π and e are such numbers. Almost all complex numbers are transcendental. Using these notions, the Gelfond–Scheider theorem states that given two algebraic numbers a and b, logb(a) is either a transcendental number or a rational number p / q (in which case aq = bp, so a and b were closely related to begin with).[44]

Calculation

Logarithms are easy to compute in some cases, such as log10(1,000) = 3 . In general, logarithms can be calculated using power series or the arithmetic-geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision.[45][46] Newton's method, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.[47] Using look-up tables, CORDIC-like methods can be used to compute logarithms if the only available operations are addition and bit shifts.[48][49] Moreover, the binary logarithm algorithm calculates lb(x) recursively based on repeated squarings of x, taking advantage of the relation

Power series

Taylor series
File:Taylor approximation of natural logarithm.gif

The Taylor series of ln(z) centered at z = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations will not converge beyond a distance of 1 from the center.

For any real number z that satisfies 0 < z < 2, the following formula holds:[nb 5][50]

This is a shorthand for saying that ln(z) can be approximated to a more and more accurate value by the following expressions:

For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465. This series approximates ln(z) with arbitrary precision, provided the number of summands is large enough. In elementary calculus, ln(z) is therefore the limit of this series. It is the Taylor series of the natural logarithm at z = 1 . The Taylor series of ln z provides a particularly useful approximation to ln(1+z) when z is small, |z| << 1, since then

For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.

More efficient series

Another series is based on the area hyperbolic tangent function:

for any real number z > 0.[nb 6][50] Using the Sigma notation, this is also written as

This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if z is close to 1. For example, for z = 1.5 , the first three terms of the second series approximate ln(1.5) with an error of about 3×10−6. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting

the logarithm of z is:

The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10b, so that ln(z) = ln(a) + b · ln(10).

A closely related method can be used to compute the logarithm of integers. From the above series, it follows that:

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1).

Arithmetic-geometric mean approximation

The arithmetic-geometric mean yields high precision approximations of the natural logarithm. ln(x) is approximated to a precision of 2p (or p precise bits) by the following formula (due to Carl Friedrich Gauss):[51][52]

Here M denotes the arithmetic-geometric mean. It is obtained by repeatedly calculating the average (arithmetic mean) and the square root of the product of two numbers (geometric mean). Moreover, m is chosen such that

Both the arithmetic-geometric mean and the constants π and ln(2) can be calculated with quickly converging series.

Applications

File:NautilusCutawayLogarithmicSpiral.jpg

A nautilus displaying a logarithmic spiral

Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral.[53] Benford's law on the distribution of leading digits can also be explained by scale invariance.[54] Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.[55] The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(x) grows very slowly for large x, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.

Logarithmic scale

Main article: Logarithmic scale
File:GermanyHyperChart.jpg

A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920s

Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a logarithmic unit of measurement. It is based on the common logarithm of ratios—10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals,[56] to describe power levels of sounds in acoustics,[57] and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels.[58] In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.[59]

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter scale. For example, a 5.0 earthquake releases 10 times and a 6.0 releases 100 times the energy of a 4.0.[60] Another logarithmic scale is apparent magnitude. It measures the brightness of stars logarithmically.[61] Yet another example is pH in chemistry; pH is the negative of the common logarithm of the activity of hydronium ions (the form hydrogen ions H+ take in water).[62] The activity of hydronium ions in neutral water is 10−7 mol·L−1, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about 10−3 mol·L−1.

Semilog (log-linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form f(x) = a · bx

appear as straight lines with slope equal to the logarithm of b.

Log-log graphs scale both axes logarithmically, which causes functions of the form f(x) = a · xk

to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.[63]

Psychology

Logarithms occur in several laws describing human perception:[64][65] Hick's law proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have.[66] Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target.[67] In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying.[68] (This "law", however, is less precise than more recent models, such as the Stevens' power law.[69])

Psychological studies found that mathematically unsophisticated individuals tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 20 as 100 is to 200. Increasing mathematical understanding shifts this to a linear estimate (positioning 100 10x as far away).[70][71]

Probability theory and statistics

File:Some log-normal distributions.svg

Three probability density functions (PDF) of random variables with log-normal distributions. The location parameter μ, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.

File:Benfords law illustrated by world's countries population.png

Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law.

Logarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.[72]

Logarithms also occur in log-normal distributions. When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.[73] Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.[74]

Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that needs to be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.[75]

Benford's law describes the occurrence of digits in many data sets, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals log10(d + 1) − log10(d), regardless of the unit of measurement.[76] Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.[77]

Computational complexity

Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem).[78] Logarithms are valuable for describing algorithms which divide a problem into smaller ones, and join the solutions of the subproblems.[79]

For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log2(N) comparisons, where N is the list's length.[80] Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to N · log(N).[81] The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor, is usually disregarded in the analysis of algorithms under the standard uniform cost model.[82]

A function f(x) is said to grow logarithmically if f(x) is (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function.[83]) For example, any natural number N can be represented in binary form in no more than log2(N) + 1 bits. In other words, the amount of memory needed to store N grows logarithmically with N.

Entropy and chaos

File:Chaotic Bunimovich stadium.png

Billiards on an oval billiard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflections at the boundary.

Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy S of some physical system is defined as

The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, pi is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2(N) bits.[84]

Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way because small errors of measurement of the initial state will predictably lead to largely different final states.[85] At least one Lyapunov exponent of a deterministically chaotic system is positive.

Fractals

File:Sierpinski dimension.svg

The Sierpinski triangle (at the right) is constructed by repeatedly replacing equilateral triangles by three smaller ones.

Logarithms occur in definitions of the dimension of fractals.[86] Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This causes the Hausdorff dimension of this structure to be log(3)/log(2) ≈ 1.58 . Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

Music

Logarithms are related to musical tones and intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. For example, the note A has a frequency of 440 Hz and B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree:

Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-21/12 logarithm of the frequency ratio, while the base-21/1200 logarithm of the frequency ratio expresses the interval in cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.[87]

Interval
(the two tones are played at the same time)
1/12 tone play  Semitone play  Just major third play  Major third play  Tritone play  Octave play 
Frequency ratio r
Corresponding number of semitones
Corresponding number of cents

Number theory

Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer x, the quantity of prime numbers less than or equal to x is denoted π(x). The prime number theorem asserts that π(x) is approximately given by

in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity.[88] As a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely proportional to the numbers of decimal digits of x. A far better estimate of π(x) is given by the offset logarithmic integral function Li(x), defined by

The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x).[89] The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.

The logarithm of n factorial, n! = 1 · 2 · ... · n , is given by

This can be used to obtain Stirling's formula, an approximation of n! for large n.[90]

Generalizations

Complex logarithm

Main article: Complex logarithm
File:Complex number illustration multiple arguments.svg

Polar form of z = x + iy. Both φ and φ' are arguments of z.

The complex numbers a solving the equation

are called complex logarithms. Here, z is a complex number. A complex number is commonly represented as z = x + iy , where x and y are real numbers and i is the imaginary unit. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number z by its absolute value, that is, the distance r to the origin, and an angle between the x axis and the line passing through the origin and z. This angle is called the argument of z. The absolute value r of z is

The argument is not uniquely specified by z: both φ and φ' = φ + 2π are arguments of z because adding 2π radians or 360 degrees[nb 7] to φ corresponds to "winding" around the origin counter-clock-wise by a turn. The resulting complex number is again z, as illustrated at the right. However, exactly one argument φ satisfies −π < φ and φ ≤ π. It is called the principal argument, denoted Arg(z), with a capital A.[91] (An alternative normalization is 0 ≤ Arg(z) < 2π.[92])

File:Complex log.jpg

The principal branch of the complex logarithm, Log(z). The black point at z = 1 corresponds to absolute value zero and brighter (more saturated) colors refer to bigger absolute values. The hue of the color encodes the argument of Log(z).

Using trigonometric functions sine and cosine, or the complex exponential, respectively, r and φ are such that the following identities hold:[93]

This implies that the a-th power of e equals z, where

φ is the principal argument Arg(z) and n is an arbitrary integer. Any such a is called a complex logarithm of z. There are infinitely many of them, in contrast to the uniquely defined real logarithm. If n = 0 , a is called the principal value of the logarithm, denoted Log(z). The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.[94]

The illustration at the right depicts Log(z). The discontinuity, that is, the jump in the hue at the negative part of the x- or real axis, is caused by the jump of the principal argument there. This locus is called a branch cut. This behavior can only be circumvented by dropping the range restriction on φ. Then the argument of z and, consequently, its logarithm become multi-valued functions.

Inverses of other exponential functions

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential.[95] Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case.[96] In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.[97]

In the context of finite groups exponentiation is given by repeatedly multiplying one group element b with itself. The discrete logarithm is the integer n solving the equation

where x is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels.[98] Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.[99]

Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of f(w) = wew,[100] and of the logistic function, respectively.[101]

Related concepts

From the perspective of pure mathematics, the identity log(cd) = log(c) + log(d) expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.[102] By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals.[103] In complex analysis and algebraic geometry, differential forms of the form df/f

are known as forms with logarithmic poles.[104]

The polylogarithm is the function defined by

It is related to the natural logarithm by Li1(z) = −ln(1 − z) . Moreover, Lis(1) equals the Riemann zeta function ζ(s).[105]

See also

  • Index of logarithm articles
  • Exponential function

Notes

  1. For further details, including the formula bm + n = bm · bn, see exponentiation or [1] for an elementary treatise.
  2. The restrictions on x and b are explained in the section "Analytic properties".
  3. Some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish ln notation," which he said no mathematician had ever used.[10] The notation was invented by Irving Stringham, a mathematician.[11][12]
  4. For example C, Java, Haskell, and BASIC.
  5. The same series holds for the principal value of the complex logarithm for complex numbers z satisfying |z − 1| < 1.
  6. The same series holds for the principal value of the complex logarithm for complex numbers z with positive real part.
  7. See radian for the conversion between 2π and 360 degrees.

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