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[[Image:Logarithms.png|thumb|315px|Logarithms to various bases: red is to base [[E (mathematical constant)|''e'']], green is to base 10, and purple is to base 1.7. Note how logarithms of all bases pass through the point (1, 0).]]

In [[mathematics]], a '''logarithm''' is a [[function (mathematics)|function]] that gives the [[Wiktionary:exponent|exponent]] in the equation ''b''<sup>''n''</sup> = ''x''. It is usually written as log<sub>''b''</sub> ''x'' = ''n''. For example:
: <math>\!\, 3^4 = 81 \mbox{, thus } \log_3 81 = 4 </math>

The logarithm is one of three closely related functions. With ''b''<sup>''n''</sup> = ''x'', ''b'' can be determined with [[radical (mathematics)|radical]]s, ''n'' with logarithms, and ''x'' with [[exponential function|exponentials]].

The '''negative of a logarithm''' is written as ''n'' = &minus;log<sub>''b''</sub> ''x''; an example of its use is in chemistry, where it expresses the [[concentration]] of hydrogen ions ([[pH]]).

An '''antilogarithm''' is used to show the inverse of the logarithm. It is written antilog<sub>''b''</sub>(''n'') and means the same as ''b''<sup>''n''</sup>.

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''.

A '''[[discrete logarithm]]''' is a related notion in the theory of finite [[group (mathematics)|groups]]. For some [[finite group]]s, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in [[cryptography]].

== Logarithms and exponentials: inverses ==
For each base (''b'' in ''b''<sup>''n''</sup>), there is one logarithm function and one exponential function; they are [[inverse function]]s. With ''b''<sup>''n''</sup> = ''x'':

* Exponentials determine ''x'' when given ''n''; to find ''x'', they multiply 1 by ''b'' as many times as (''n'').
* Logarithms determine ''n'' when given ''x''; ''n'' is the number of times that ''x'' must be divided by ''b'' to reach 1.

== Using logarithms ==
The function log<sub>''b''</sub>(''x'') is defined whenever ''x'' is a [[negative and non-negative numbers|positive]] [[real number]] and ''b'' is a positive real number different from 1. See [[logarithmic identities]] for several rules governing the logarithm functions. Logarithms may also be defined for [[complex number|complex]] arguments. This is explained on the [[natural logarithm]] page.

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 (and in particular if they are [[coprime]] and both greater than 1). 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.

=== Unspecified bases ===

* Mathematicians generally understand either "ln(''x'')" or "log(''x'')" to mean log<sub>e</sub>(''x''), i.e., the natural logarithm of ''x'', and write "log<sub>10</sub>(''x'')" if the base-10 logarithm of ''x'' is intended.

* Engineers, biologists, and some others write only "ln(''x'')" or (occasionally) "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, in the context of [[computing]], [[binary logarithm|log<sub>2</sub>]](''x'').

* Sometimes Log(''x'') (capital ''L'') is used to mean log<sub>10</sub>(''x''), by those people who use log(''x'') with a lowercase ''l'' to mean log<sub>''e''</sub>(''x'').

* The notation Log(''x'') is also used by mathematicians to denote the [[principal branch]] of the (natural) logarithm function.

* In most commonly used [[programming language]]s, including [[C programming language|C]], [[C plus plus|C++]], <!--[[Pascal programming language|Pascal]], -->[[Fortran]], and [[BASIC programming language|BASIC]], "log" or "LOG" means natural logarithm.

Most of the reason for thinking about base-10 logarithms became obsolete after the early 1970s when hand-held calculators became widespread (for more on this point, see [[common logarithm]]). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(''x'')" to mean the base-10 logarithm of ''x'' and use only "ln(''x'')" to refer to the natural logarithm of ''x''. As recently as 1984, [[Paul Halmos]] in his autobiography 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]], some mathematicians have adopted the "ln" notation, but most use "log". In computer science, the base 2 logarithm is written as lg(''x'') to avoid confusion. This usage was suggested by Edward Reingold and popularized by [[Donald Knuth]].

When "log" is written without a base (''b'' missing from log<sub>''b''</sub>), the intent can usually be determined from context:

* [[natural logarithm]] (log<sub>''[[e (mathematical constant)|e]]</sub>'') in [[analysis (mathematics)|mathematical analysis]];
* [[binary logarithm]] (log<sub>2</sub>) with [[interval (music)|musical interval]]s and in subjects that deal with [[bit]]s;
* [[common logarithm]] (log<sub>10</sub>) when logarithm tables are used to simplify hand calculations;
* [[indefinite logarithm]] when the base is irrelevant.

=== Change of base ===
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'':
: <math> \log_b(x) = \frac{\log_k(x)}{\log_k(b)} </math>

{| align="center" cellpadding="3" style="border:1px solid"
|+ '''Base change formula proof'''
| <math>b^{\log_b(x)} = x\!\,</math> || by definition
|-
| <math>\log_k\left( b^{\log_b(x)} \right) = \log_k(x)</math> || take logs of both sides
|-
| <math>\log_b(x)\, \log_k(b) = \log_k(x)</math> || simplify the left hand side
|-
| <math>\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\,\!</math> || divide by log<sub>''k''</sub>(''b'')
|}

All this implies, moreover, that all logarithm functions (whatever the base) are [[similar]] to each other.

== Uses of logarithms ==
Logarithms are useful in solving equations in which exponents are unknown. They have simple [[derivative]]s, so they are often used as the solution of [[integral]]s. Furthermore, various quantities in science are expressed as logarithms of other quantities; see [[logarithmic scale]] for an explanation and a list.

=== Exponential functions ===
Sometimes (especially in the context of analysis) it is necessary to calculate arbitrary exponential functions <math>f(x)^x</math> using only the [[exponential function|natural exponent]] <math>e^x</math>:

<math>f(x)^x = e^{\log(f(x)^x)}</math><br>
<math>= e^{x\log(f(x))}</math>

=== Easier computations ===
Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes a few operations easier:
{| 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}) = \log(a) / b </math>
|}
Before [[calculator|electronic calculator]]s, this made difficult operations on two numbers much easier. One simply found the logarithms of both numbers (multiply and divide) or the first number (power or root, where one number is already an exponent) in a table of [[common logarithm]]s, performed a simpler operation on those, and found the result on a table. [[Slide rule]]s performed the same operations using logarithms, but faster and with higher 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]]).

In [[abstract algebra]], this property of the logarithm functions can be summarized by noting that any logarithm function with a fixed base is a [[group isomorphism]] from the [[group (mathematics)|group]] of strictly positive [[real number]]s under multiplication to the group of all real numbers under addition.

=== Calculus ===
To calculate the [[derivative]] of a logarithmic function, the following formula is used
: <math>\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}</math>
where ''ln'' is the natural logarithm, i.e. with base ''e''. Letting ''b'' = ''e'':
: <math>\frac{d}{dx} \ln(x) = \frac{1}{x}, \qquad \int \frac{1}{x} \,dx = \ln(x) + C</math>

One can then see that the following formula gives the [[integral]] of a logarithm
: <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>

''See also:'' [[Wikisource:Table of common limits#Logarithmic and exponential functions|table of limits of logarithmic functions]], [[list of integrals of logarithmic functions]]

== History ==
[[Joost Bürgi]], a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first conceived of logarithms. The method of natural logarithms was first 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 invention. 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.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word ''logarithm'', a [[portmanteau]], to mean a number that indicates a ratio: &lambda;o&gamma;o&sigmaf; (''logos'') meaning ratio, and &alpha;&rho;&iota;&theta;&mu;o&sigmaf; (''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.

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>, <math>L</math> is approximately <math>10^7\log_{1/e} N/10^7</math>.

=== Tables of logarithms ===
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.

In [[1617]], 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 [[Adrian Vlacq]], a [[the Netherlands|Dutch]] computer; but in his table, which appeared in [[1628]], the logarithms were given to only ten places of decimals.

Vlacq's table was later to 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]].

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.

Briggs and Vlacq also published original tables of the logarithms of the [[trigonometric function]]s.

Besides the tables mentioned above, a great collection, called ''Tables du Cadastre,'' was constructed under the direction of [[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.

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.

== Algorithm ==
To calculate log<sub>''b''</sub>(''x'') if ''b'' and ''x'' are [[rational numbers]] and ''x'' &ge; ''b'' &gt; 1:

If ''n<sub>0</sub>'' is the largest [[natural number]] such that ''b''<sup>''n<sub>0</sub>''</sup> &le; ''x'' or, alternately,
: <math> n_0 = \lfloor \log_b(x) \rfloor </math>
then
: <math> \log_b(x) = n_0 + \frac{1}{\log_{x / b^{n_0}}(b)} </math>

This [[algorithm]] [[recursion|recursively]] produces the [[continued fraction]]
: <math> \log_b(x) = n_0 + \frac{1}{n_1 + \frac{1}{n_2 + \frac{1}{n_3 + \cdots}}}. </math>

The logarithms produced are [[irrational number|irrational]] for most inputs.

To use [[irrational number]]s as inputs, apply the algorithm to successively detailed rational approximations. The limit of the result [[series]] should converge to the actual result.

:<math>\ln 2 = 2\sum_{n = 0}^{\infty }\frac{(1/5)^{
2\,n+1}}{ 2\,n+1}+2\sum_{n = 0}^{\infty }\frac{(1/7)^{
2\,n+1}}{ 2\,n+1}</math>

{| align="center" style="border:1px solid" cellpadding="3"
|+ '''Algorithm proof'''
| <math> \log_b(x) = \log_b(x)\,\! </math> || identity
|-
| <math> \log_b(x) = n_0 + \log_b(x) - n_0\,\! </math> || algebraic manipulation
|-
| <math> \log_b(x) = n_0 + \log_b(x) - \log_b(b^{n_0})\,\! </math> || [[logarithmic identities|logarithmic identity]]
|-
| <math> \log_b(x) = n_0 + \log_b\left(\begin{matrix}\frac{x}{b^{n_0}}\end{matrix}\right) </math> || logarithmic identity
|-
| <math> \log_b(x) = n_0 + \frac{1}{\log_{\begin{matrix}\frac{x}{b^{n_0}}\end{matrix}}(b)} </math> || base switch
|}

== Trivia ==
=== Unicode glyph ===
''log'' has its own [[Unicode]] [[glyph]]: &#13266; (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.

=== Alternate notation ===
A few people use the notation <sup>''b''</sup>log(''x'') instead of log<sub>''b''</sub>(''x'').

=== Relationships between binary, natural and common logarithms ===
In particular we have:
: log<sub>2</sub>(''e'') &asymp; 1.44269504
: log<sub>2</sub>(10) &asymp; 3.32192809
: log<sub>''e''</sub>(10) &asymp; 2.30258509
: log<sub>''e''</sub>(2) &asymp; 0.693147181
: log<sub>10</sub>(2) &asymp; 0.301029996
: log<sub>10</sub>(''e'') &asymp; 0.434294482

A curious coincidence is the [[approximation]] log<sub>2</sub>(''x'') &asymp; 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> &asymp; 1 (in fact 1.0084...). The property is demonstrated in all six conversion factors above, arranged in pairs of two:
{| cellpadding=3
|-
| 2.30 || 3.32
|-
| 0.30 || 0.69
|-
| 0.43 || 1.44
|}
This comes on top of the reciprocal relations we have:
{| cellpadding=3
|-
| 2.30 || 0.43
|-
| 0.30 || 3.32
|-
| 0.69 || 1.44
|}
Another interesting coincidence is that log<sub>10</sub>(2) &asymp; 0.3 (the actual value is about 0.301029995); this corresponds to the fact that, with an error of only 2.4%, 2<sup>10</sup> &asymp; 10<sup>3</sup>
(i.e. 1024 is about 1000; see also [[Binary prefix]]).

== See also ==
* [[Logarithmic identities]]
* [[Natural logarithm]]
* [[Common logarithm]]
* [[Indefinite logarithm]]
* [[Logarithmic units]]
* [[Discrete logarithm]]
* [[Zech's logarithms]]
* [[Logarithm of a matrix]]

== References ==
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]].

== External links ==
* [http://mathworld.wolfram.com/Logarithm.html Logarithm] on [[MathWorld]]
* [http://www.micheloud.com/FXM/LOG/index.htm Jost Burgi, Swiss Inventor of Logarithms]
* [http://www.algebra.com/calculators/algebra/logarithm/ Logarithm calculators and word problems with work shown, for school students]
* [http://johnnapier.com/table_of_logarithms_001.htm/ Translation of Napier's work on logarithms]

[[Category:Mathematical analysis]]
[[Category:Special functions]]
[[Category:Portmanteaus]]

[[bg:Логаритъм]]
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[[ja:対数]]
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[[sl:Logaritem]]
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[[sv:Logaritm]]
[[zh:对数]]

Revision as of 00:07, 12 November 2005

NIGGA STOLE MY BIKE! 849696659

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