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| {{redirect|Base e|the numbering system which uses [[e (mathematical constant)|e]] as its base|Non-integer representation#Base e}}
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| [[Image:Log.svg|thumb|right|300 px|Graph of the natural logarithm function. The function slowly grows to positive infinity as ''x'' increases and rapidly goes to negative infinity as ''x'' approaches 0 ("slowly" and "rapidly" as compared to any [[power law]] of ''x''); the y-axis is an [[asymptote]].]]
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| The '''natural logarithm''' is the [[logarithm]] to the [[base (exponentiation)|base]] ''[[e (mathematical constant)|e]]'', where ''e'' is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] constant approximately equal to 2.718<span style="margin-left:0.25em">281</span><span style="margin-left:0.2em">828</span>. The natural logarithm is generally written as ln ''x'', log<sub>''e''</sub> ''x'' or sometimes, if the base of ''e'' is implicit, as simply log ''x''.<ref>{{cite book
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| |title=Mathematics for physical chemistry
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| |edition=3rd
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| |first1=Robert G.
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| |last1=Mortimer
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| |publisher=Academic Press
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| |year=2005
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| |isbn=0-12-508347-5
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| |page=9
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| |url=http://books.google.com/books?id=nGoSv5tmATsC}}, [http://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]
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| </ref> Parentheses are sometimes added for clarity, giving ln(''x''), log<sub>''e''</sub>(''x'') or log(''x''). This is done in particular when the argument to the logarithm is not a single symbol, in order to prevent ambiguity.
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| The natural logarithm of a number ''x'' is the [[exponentiation|power]] to which ''e'' would have to be raised to equal ''x''. For example, ln(7.389...) is 2, because ''e''<sup>2</sup>=7.389.... The natural log of ''e'' itself, ln(''e''), is 1 because ''e''<sup>1</sup> = ''e'', while the natural logarithm of 1, ln(1), is 0, since ''e''<sup>0</sup> = 1. | |
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| The natural logarithm can be defined for any positive [[real number]] ''a'' as the [[Integral|area under the curve]] ''y'' = 1/''x'' from 1 to ''a''. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural." The definition can be extended to non-zero [[complex number]]s, as explained below.
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| The natural logarithm function, if considered as a real-valued function of a real variable, is the [[inverse function]] of the [[exponential function]], leading to the identities:
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| :<math>e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!</math>
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| :<math>\ln(e^x) = x.\,\! </math>
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| Like all logarithms, the natural logarithm maps multiplication into addition:
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| :<math> \ln(xy) = \ln(x) + \ln(y). \!\, </math>
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| Thus, the logarithm function is an [[isomorphism]] from the [[group (mathematics)|group]] of positive real numbers under multiplication to the group of real numbers under addition, represented as a [[function (mathematics)|function]]:
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| :<math>\ln : \mathbb{R}^+ \to \mathbb{R}.</math>
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| Logarithms can be defined to any positive base other than 1, not just ''e''. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. For instance, the [[binary logarithm]] is just the natural logarithm divided by ln(2), the [[natural logarithm of 2]]. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in [[exponential decay]] problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving [[compound interest]].
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| {| class=infobox width=200px
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| |colspan=2 align=center| Natural Logarithm
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| |-
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| |'''Representation''' ||<math>\ln x \,</math>
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| |-
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| |'''Inverse''' ||<math>e^x \,</math>
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| |-
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| |'''Derivative''' ||<math>\frac{1}{x} \,</math>
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| |-
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| |'''Indefinite Integral''' ||<math>x\ln x - x + C \,</math>
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| |}
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| {{E (mathematical constant)}}
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| ==History==
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| The first mention of the natural logarithm was by [[Nicholas Mercator]] in his work ''Logarithmotechnia'' published in 1668,<ref>{{cite web |author=J J O'Connor and 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=September 2001 |accessdate=2009-02-02}}</ref> although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm.<ref>{{cite book |last=Cajori |first=Florian |authorlink=Florian Cajori |title=A History of Mathematics, 5th ed |pages=152 |publisher=AMS Bookstore |year=1991 |isbn=0-8218-2102-4 |url=http://books.google.com/?id=mGJRjIC9fZgC&dq=%22Cajori%22+%22A+History+of+Mathematics%22+}}</ref> It was formerly also called hyperbolic logarithm,<ref>{{cite web |author=Flashman, Martin |url=http://users.humboldt.edu/flashman/Presentations/Estimations.html |title=Estimating Integrals using Polynomials |accessdate=2008-03-23}}</ref> as it corresponds to the area under a [[hyperbola]]. It is also sometimes referred to as the [[Napierian logarithm]], although the original meaning of this term is slightly different.
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| ==Notational conventions==
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| The notations {{nowrap|"ln ''x''"}} and {{nowrap|"log<sub>''e''</sub> ''x''"}} both refer unambiguously to the natural logarithm of ''x''.
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| {{nowrap|"log ''x''"}} without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many [[programming language]]s.<ref>Including [[C (programming language)|C]], [[C++]], [[SAS System|SAS]], [[MATLAB]], [[Mathematica]], <!--[[Pascal programming language|Pascal]], -->[[Fortran]], and [[BASIC programming language|BASIC]]</ref> {{nowrap|"log ''x''"}} is frequently used to denote the [[common logarithm|common (base 10) logarithm]], however.
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| ==Origin of the term ''natural logarithm''==
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| Initially, it might seem that since the common numbering system is [[base 10]], this base would be more "natural" than base ''e''. But mathematically, the number 10 is not particularly significant. Its use culturally—as the basis for many societies’ numbering systems—likely arises from humans’ typical number of fingers.<ref>{{cite book | last=Boyers | first=Carl | title=A History of Mathematics | publisher=[[John Wiley & Sons|Wiley]] | year=1968}}</ref> Other cultures have based their [[counting systems]] on such choices as [[quinary|5]], [[octal|8]], [[duodecimal|12]], [[vigesimal|20]], and [[sexagesimal|60]].<ref>{{cite journal | last=Harris | first=John | title=Australian Aboriginal and Islander mathematics | journal=Australian Aboriginal Studies | volume=2 |pages=29–37 | year=1987 | url=http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0005975_v_a.pdf| accessdate=2008-02-12 | format=PDF}}</ref><ref>{{cite journal | last=Large| first=J.J. | title=The vigesimal system of enumeration | journal=Journal of the Polynesian Society| volume=11|issue=4 |pages=260–261 | year=1902 | url=http://www.jps.auckland.ac.nz/document/?wid=636 | accessdate=30 March 2011}}</ref><ref>{{cite journal | last=Cajori| first=Florian| authorlink = Florian Cajori | title=Sexagesimal fractions among the Babylonians | journal=American Mathematical Monthly| volume=29 |pages=8–10 | year=1922| doi=10.2307/2972914 | jstor=2972914 | issue=1 }}</ref>
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| log<sub>''e''</sub> is a "natural" log because it automatically springs from, and appears so often in, mathematics. For example, consider the problem of [[derivative|differentiating]] a logarithmic function:<ref>{{cite book
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| |title=Calculus: An Applied Approach
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| |edition=8th
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| |first1=Ron
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| |last1=Larson
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| |publisher=Cengage Learning
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| |year=2007
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| |isbn=0-618-95825-8
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| |page=331
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| |url=http://books.google.com/books?id=rbDG7V0OV34C}}, [http://books.google.com/books?id=rbDG7V0OV34C&pg=PA331 Section 4.5, page 331]
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| </ref>
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| :<math>\frac{d}{dx}\log_b(x) = \frac{d}{dx} \left( \frac{1}{\ln(b)} \ln{x} \right) = \frac{1}{\ln(b)} \frac{d}{dx} \ln{x} = \frac{1}{x\ln(b)}. </math>
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| If the [[base (exponentiation)|base]] ''b'' equals ''e'', then the derivative is simply 1/''x'', and at ''x'' = 1 this derivative equals 1. Another sense in which the base-''e''-logarithm is the most natural is that it can be defined quite easily in terms of a simple integral or [[Taylor series]] and this is not true of other logarithms.
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| Further senses of this naturalness make no use of [[calculus]]. As an example, there are a number of simple series involving the natural logarithm. [[Pietro Mengoli]] and [[Nicholas Mercator]] called it ''logarithmus naturalis'' a few decades before [[Sir Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]] developed calculus.<ref>{{cite web | last=Ballew | first=Pat | title=Math Words, and Some Other Words, of Interest | url=http://www.pballew.net/arithme1.html#ln | accessdate=2007-09-16}}</ref>
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| == Definitions ==
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| [[Image:Log-pole-x 1.svg|thumb|ln(''a'') illustrated as the area under the curve ''f''(''x'') = 1/''x'' from 1 to ''a''. If ''a'' is less than 1, the area from ''a'' to 1 is counted as negative.]]
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| Formally, ln(''a'') may be defined as the [[integral]],
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| :<math>\ln(a)=\int_1^a \frac{1}{x}\,dx.</math>
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| This function is a logarithm because it satisfies the fundamental property of a logarithm:
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| :<math>\ln(ab)=\ln(a)+\ln(b). \,\!</math>
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| This can be demonstrated by splitting the integral that defines ln(''ab'') into two parts and then making the [[Integration by substitution|variable substitution]] {{nowrap|1=''x'' = ''ta''}} in the second part, as follows:
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| :<math>
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| \ln (ab)
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| = \int_1^{ab} \frac{1}{x} \; dx
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| = \int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx
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| =\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{at} \; d(at)
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| </math>
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| ::<math>
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| =\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt
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| = \ln (a) + \ln (b).
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| </math>
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| The number ''[[E (mathematical constant)|e]]'' can then be defined as the unique real number ''a'' such that ln(''a'') = 1.
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| Alternatively, if the [[exponential function]] has been defined first, say by using an [[infinite series]], the natural logarithm may be defined as its [[inverse function]], i.e., ln is that function such that exp(ln(''x'')) = ''x''. Since the range of the exponential function on real arguments is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive ''x''.
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| == Properties ==
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| * <math>\ln(1) = 0 \,</math>
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| * <math>\ln(-1) = i \pi \,</math>
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| ::(see [[complex logarithm]])
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| * <math>\ln(x) < \ln(y) \quad{\rm for}\quad 0 < x < y \,</math>
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| * <math>\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \,</math>
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| * <math>\ln(x^y) = y \, \ln(x) \,</math>
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| * <math>\frac{x-1}{x} \leq \ln(x) \leq x-1 \quad{\rm for}\quad x > 0 \,</math>
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| * <math>\ln{( 1+x^\alpha )} \leq \alpha x \quad{\rm for}\quad x \ge 0, \alpha \ge 1 \,</math>
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| :{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
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| !Proof
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| The statement is true for <math>x=0</math>, and we now show that <math>\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>, which completes the proof by the [[fundamental theorem of calculus#First part|fundamental theorem of calculus]]. Hence, we want to show that
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| :<math>\frac{d}{dx} \ln{( 1+x^\alpha )} = \frac{\alpha x^{\alpha - 1}}{1 + x^\alpha} \leq \alpha = \frac{d}{dx} ( \alpha x ) </math>
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| (Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity <math>(1+x^\alpha) / \alpha</math> and subtracting <math>x^\alpha</math> we would obtain
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| :<math> x^{\alpha-1} \leq x^\alpha + 1 </math>
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| :<math> x^{\alpha-1} (1-x) \leq 1 </math>
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| This statement is trivially true for <math>x \ge 1</math> since the left hand side is negative or zero. For <math>0 \le x < 1</math> it is still true since both factors on the left are less than 1 (recall that <math>\alpha \ge 1</math>). Thus this last statement is true and by repeating our steps in reverse order we find that <math>\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>. This completes the proof.
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| |}
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| == Derivative, Taylor series ==
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| [[Image:LogTay.svg|300px|thumb|right|The Taylor polynomials for ln(1 + ''x'') only provide accurate approximations in the range −1 < ''x'' ≤ 1. Note that, for ''x'' > 1, the Taylor polynomials of higher degree are ''worse'' approximations.]]
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| The [[derivative]] of the natural logarithm is given by | |
| :<math>\frac{d}{dx} \ln(x) = \frac{1}{x}.\,</math>
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| This leads to the [[Taylor series]] for ln(1 + ''x'') around 0; also known as the [[Mercator series]]
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| :<math>\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad{\rm for}\quad \left|x\right| \leq 1,\quad</math>
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| ::<math>{\rm unless}\quad x = -1.</math>
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| ([[Leonhard Euler]]<ref>[[Leonhard Euler]], Introductio in Analysin Infinitorum. Tomus Primus. Bousquet, Lausanne 1748. Exemplum 1, p. 228; quoque in: Opera Omnia, Series Prima, Opera Mathematica, Volumen Octavum, Teubner 1922</ref> nevertheless boldly applied this series to ''x= -1'',
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| in order to show that the harmonic series equals the (natural) logarithm of ''1/(1-1)'', that is the logarithm of infinity. Nowadays, more formally but perhaps less vividly, we prove that the harmonic series truncated at ''N'' is close to the logarithm of ''N'', when ''N'' is large).
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| At right is a picture of ln(1 + ''x'') and some of its [[Taylor polynomial]]s around 0. These approximations converge to the function only in the region −1 < ''x'' ≤ 1; outside of this region the higher-degree Taylor polynomials are ''worse'' approximations for the function.
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| Substituting ''x'' − 1 for ''x'', we obtain an alternative form for ln(x) itself, namely
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| :<math>\ln(x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} (x-1) ^ n</math>
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| :<math>\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots</math>
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| :<math>{\rm for}\quad \left|x-1\right| \leq 1\quad {\rm unless}\quad x = 0 \,.</math><ref>[http://www.math2.org/math/expansion/log.htm "Logarithmic Expansions" at Math2.org]</ref>
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| By using the [[binomial transform#Euler transform|Euler transform]] on the Mercator series, one obtains the following, which is valid for any x with absolute value greater than 1:
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| :<math>\ln{x \over {x-1}} = \sum_{n=1}^\infty {1 \over {n x^n}} = {1 \over x}+ {1 \over {2x^2}} + {1 \over {3x^3}} + \cdots \,.</math>
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| This series is similar to a [[BBP-type formula]].
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| Also note that <math> x \over {x-1} </math> is its own inverse function, so to yield the natural logarithm of a certain number ''y'', simply put in <math> y \over {y-1} </math> for ''x''.
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| :<math>\ln{x} = \sum_{n=1}^\infty {1 \over {n}} \left( {x - 1 \over x} \right)^n = \left( {x - 1 \over x} \right) + {1 \over 2} \left( {x - 1 \over x} \right)^2 + {1 \over 3} \left( {x - 1 \over x} \right)^3 + \cdots \,</math>
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| :<math>{\rm for }\quad \operatorname{Re} (x) \geq \frac12 \,.</math>
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| == The natural logarithm in integration ==
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| The natural logarithm allows simple [[integral|integration]] of functions of the form ''g''(''x'') = ''f'' '(''x'')/''f''(''x''): an [[antiderivative]] of ''g''(''x'') is given by ln(|''f''(''x'')|). This is the case because of the [[chain rule]] and the following fact:
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| :<math>\ {d \over dx}\left( \ln \left| x \right| \right) = {1 \over x}.</math>
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| In other words,
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| :<math>\int { 1 \over x} dx = \ln|x| + C</math>
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| and
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| :<math>\int { \frac{f'(x)}{f(x)}\, dx} = \ln |f(x)| + C.</math>
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| Here is an example in the case of ''g''(''x'') = tan(''x''):
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| :<math>\int \tan (x) \,dx = \int {\sin (x) \over \cos (x)} \,dx</math>
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| :<math>\int \tan (x) \,dx = \int {-{d \over dx} \cos (x) \over {\cos (x)}} \,dx.</math>
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| Letting ''f''(''x'') = cos(''x'') and ''f'''(''x'')= – sin(''x''):
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| :<math>\int \tan (x) \,dx = -\ln{\left| \cos (x) \right|} + C</math>
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| :<math>\int \tan (x) \,dx = \ln{\left| \sec (x) \right|} + C</math>
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| where ''C'' is an [[arbitrary constant of integration]].
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| The natural logarithm can be integrated using [[integration by parts]]:
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| :<math>\int \ln (x) \,dx = x \ln (x) - x + C.</math>
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| == Numerical value ==<!-- This section is linked from [[Common logarithm]] -->
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| To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:
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| :<math>\ln(1+x)= x \,\left( \frac{1}{1} - x\,\left(\frac{1}{2} - x \,\left(\frac{1}{3} - x \,\left(\frac{1}{4} - x \,\left(\frac{1}{5}- \cdots \right)\right)\right)\right)\right) \quad{\rm for}\quad \left|x\right|<1.\,\!</math>
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| To obtain a better rate of convergence, the following identity can be used.
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| :{|
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| |-
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| |<math>\ln(x) = \ln\left(\frac{1+y}{1-y}\right)</math>
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| |<math>= 2\,y\, \left( \frac{1}{1} + \frac{1}{3} y^{2} + \frac{1}{5} y^{4} + \frac{1}{7} y^{6} + \frac{1}{9} y^{8} + \cdots \right) </math>
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| |-
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| |<math>= 2\,y\, \left( \frac{1}{1} + y^{2} \, \left( \frac{1}{3} + y^{2} \, \left( \frac{1}{5} + y^{2} \, \left( \frac{1}{7} + y^{2} \, \left( \frac{1}{9} + \cdots \right) \right) \right)\right) \right) </math>
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| |}
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| :provided that ''y'' = (''x''−1)/(''x''+1) and Re(''x'') ≥ 0 but ''x'' ≠ 0.
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| For ln(''x'') where ''x'' > 1, the closer the value of ''x'' is to 1, the faster the rate of convergence. The identities associated with the logarithm can be leveraged to exploit this:
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| :{|
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| |-
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| |<math>\ln(123.456)\!</math>
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| |<math>= \ln(1.23456 \times 10^2) \,\!</math>
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| |<math>= \ln(1.23456) + \ln(10^2) \,\!</math>
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| |<math>= \ln(1.23456) + 2 \times \ln(10) \,\!</math>
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| |-
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| |<math>\approx \ln(1.23456) + 2 \times 2.3025851. \,\!</math>
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| |}
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| Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.
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| ===Natural logarithm of 10===
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| The natural logarithm of 10 ({{OEIS2C|A002392}}) plays a role for example in computation of natural logarithms of numbers represented in the [[scientific notation]], a mantissa multiplied by a power of 10:
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| : <math>\ln (a\times 10^n) = \ln a + n \ln 10.</math>
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| By this scaling, the algorithm may reduce the logarithm of all positive real numbers to an algorithm for natural logarithms in the range <math>1\le a< 10</math>.
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| ===High precision===
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| To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. If ''x'' is near 1, an alternative is to use [[Newton's method]] to invert the exponential function, whose series converges more quickly. For an optimal function, the iteration simplifies to
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| :<math> y_{n+1} = y_n + 2 \cdot \frac{ x - \exp ( y_n ) }{ x + \exp ( y_n ) } \,</math>
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| which has [[cubic convergence]] to ln(''x'').
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| Another alternative for extremely high precision calculation is the formula<ref>{{cite journal |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>
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| <ref> | |
| {{Cite journal
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| |first1=Timm
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| |last1=Ahrendt
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| |title=Stacs 99
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| |series=Lecture notes in computer science
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| |doi=10.1007/3-540-49116-3_28
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| |volume=1564
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| |year=1999
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| |pages=302–312
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| |postscript=<!--None-->
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| |chapter=Fast Computations of the Exponential Function
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| |isbn=978-3-540-65691-3
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| }}
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| </ref>
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| :<math>\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2,</math>
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| where ''M'' denotes the [[arithmetic-geometric mean]] of 1 and 4/s, and
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| :<math>s = x \,2^m > 2^{p/2},</math>
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| with ''m'' chosen so that ''p'' bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and [[pi|π]] can be pre-computed to the desired precision using any of several known quickly converging series.)
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| ===Computational complexity===
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| {{main|Computational complexity of mathematical operations}}
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| The [[computational complexity]] of computing the natural logarithm (using the arithmetic-geometric mean) is O(''M''(''n'') ln ''n''). Here ''n'' is the number of digits of precision at which the natural logarithm is to be evaluated and ''M''(''n'') is the computational complexity of multiplying two ''n''-digit numbers.
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| ==Continued fractions==
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| While no simple [[continued fraction]]s are available, several [[generalized continued fraction]]s are, including:
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| :<math>
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| \ln (1+x)=\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots=
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| \cfrac{x}{1-0x+\cfrac{1^2x}{2-1x+\cfrac{2^2x}{3-2x+\cfrac{3^2x}{4-3x+\cfrac{4^2x}{5-4x+\ddots}}}}}
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| </math>
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| :<math>
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| \ln \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\ddots}}}}}}
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| = \cfrac{2x} {2y+x-\cfrac{(1x)^2} {3(2y+x)-\cfrac{(2x)^2} {5(2y+x)-\cfrac{(3x)^2} {7(2y+x)-\ddots}}}}
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| </math>
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| These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed by repeatedly adding those of smaller numbers, with similarly rapid convergence.
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| For example, since 2 = 1.25<sup>3</sup> × 1.024, the [[natural logarithm of 2]] can be computed as:
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| :<math>
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| \ln 2 = 3 \ln \left( 1+\frac{1}{4} \right) + \ln \left( 1+\frac{3}{125} \right)
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| = \cfrac{6} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
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| + \cfrac{6} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
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| </math>
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| Furthermore, since 10 = 1.25<sup>10</sup> × 1.024<sup>3</sup>, even the natural logarithm of 10 similarly can be computed as:
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| :<math>
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| \ln 10 = 10 \ln \left( 1+\frac{1}{4} \right) + 3\ln \left( 1+\frac{3}{125} \right)
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| = \cfrac{20} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
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| + \cfrac{18} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
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| </math>
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| ==Complex logarithms==
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| {{Main|Complex logarithm}}
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| The exponential function can be extended to a function which gives a [[complex number]] as ''e''<sup>''x''</sup> for any arbitrary complex number ''x''; simply use the infinite series with ''x'' complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no ''x'' has ''e''<sup>''x''</sup> = 0; and it turns out that ''e''<sup>2''πi''</sup> = 1 = ''e''<sup>0</sup>. Since the multiplicative property still works for the complex exponential function, ''e''<sup>''z''</sup> = ''e''<sup>''z''+2''nπi''</sup>, for all complex ''z'' and integers ''n''.
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| So the logarithm cannot be defined for the whole [[complex plane]], and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2''πi'' at will. The complex logarithm can only be single-valued on the [[complex plane#Cutting the plane|cut plane]]. For example, ln ''i'' = 1/2 ''πi'' or 5/2 ''πi'' or −3/2 ''πi'', etc.; and although ''i''<sup>4</sup> = 1, 4 log ''i'' can be defined as 2''πi'', or 10''πi'' or −6 ''πi'', and so on.
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| <gallery mode=packed caption="Plots of the natural logarithm function on the complex plane ([[principal branch]])">
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| Image:Natural Logarithm Re.svg| ''z'' = Re(ln(x+iy))
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| Image:Natural Logarithm Im Abs.svg| ''z'' = |Im(ln(x+iy))|
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| Image:Natural Logarithm Abs.svg| ''z'' = |ln(x+iy)|
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| Image:Natural Logarithm All.svg| Superposition of the previous 3 graphs
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| </gallery>
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| ==See also==
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| * [[John Napier]] – inventor of logarithms
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| * [[Logarithm of a matrix]]
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| * [[Logarithmic integral function]]
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| * [[Nicholas Mercator]] – first to use the term natural log
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| * [[Polylogarithm]]
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| * [[Von Mangoldt function]]
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| * [[e (mathematical constant)|The number ''e'']]
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * [http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ Demystifying the Natural Logarithm (ln) | BetterExplained]
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| {{DEFAULTSORT:Natural Logarithm}}
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| [[Category:Logarithms]]
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| [[Category:Elementary special functions]]
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| [[Category:E (mathematical constant)]]
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| [[de:Logarithmus#Natürlicher Logarithmus]]
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