View source for Natural logarithm

{{redirect|Base e|the numbering system which uses "e" as its base|Non-integer representation#Base&nbsp;e}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
[[File:Log.svg|thumb|right|300 px|Graph of the natural logarithm function. The function slowly grows to positive infinity as ''x'' increases and slowly goes to negative infinity as ''x'' approaches 0 ("slowly" as compared to any [[power law]] of ''x''); the ''y''-axis is an [[asymptote]].]]

The '''natural logarithm''' of a number is its [[logarithm]] to the [[base (exponentiation)|base]] of the [[mathematical constant]] [[e (mathematical constant)|''e'']], where ''e'' is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] number approximately equal to {{val|2.718281828459}}. The natural logarithm of ''x'' is generally written as {{nowrap|ln ''x''}}, {{nowrap|log<sub>''e''</sub> ''x''}},  or sometimes, if the base ''e'' is implicit, simply {{nowrap|log ''x''}}.<ref>{{cite book |title=Mathematics for physical chemistry |edition=3rd |author-first=Robert G. |author-last=Mortimer |publisher=[[Academic Press]] |date=2005 |isbn=0-12-508347-5 |page=9 |url=https://books.google.com/books?id=nGoSv5tmATsC}} [https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]</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, to prevent ambiguity.

The natural logarithm of ''x'' is the [[exponentiation|power]] to which ''e'' would have to be raised to equal ''x''. For example, ln(7.5) is 2.0149..., because {{nowrap|1=''e''<sup>2.0149...</sup> = 7.5}}. The natural log of ''e'' itself, ln(''e''), is 1, because {{nowrap|1=''e''<sup>1</sup> = ''e''}}, while the natural logarithm of 1, ln(1), is 0, since {{nowrap|1=''e''<sup>0</sup> = 1}}.

The natural logarithm can be defined for any positive [[real number]] ''a'' as the [[Integral|area under the curve]] {{nowrap|1=''y'' = 1/''x''}}  from 1 to ''a'' (the area being taken as negative when ''a'' < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can be extended to give logarithm values for negative numbers and for all non-zero [[complex number]]s, although this leads to a [[multi-valued function]]: see [[Complex logarithm]].

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:
:<math>e^{\ln x} = x \qquad \text{if }x > 0</math>

:<math>\ln(e^x) = x.</math>

Like all logarithms, the natural logarithm maps multiplication into addition:

:<math> \ln(xy) = \ln x + \ln y.</math>

Thus, the logarithm function is a [[group isomorphism]] from [[positive real numbers]] under multiplication to the [[group (mathematics)|group]] of real numbers under addition, represented as a [[function (mathematics)|function]]:
:<math>\ln \colon \mathbb{R}^+ \to \mathbb{R}.</math>

Logarithms can be defined to any positive base other than 1, not only ''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 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]].

By [[Lindemann–Weierstrass theorem]], the natural logarithm of any positive [[algebraic number]] other than 1 is a [[transcendental number]].

{| class=infobox width=247px
|colspan=2 align=center| Natural logarithm
|-
|'''Representation''' ||<math>\ln x</math>
|-
|'''Inverse''' ||<math>e^x</math>
|-
|'''Derivative''' ||<math>\frac{1}{x}</math>
|-
|'''nth Derivative'''
|<math>-(n-1)!\cdot\frac{(-1)^{n}}{x^n}</math>
|-
|'''Indefinite Integral''' ||<math>x\ln x - x + C</math>
|}
{{E (mathematical constant)}}

==History==
{{Main|History of logarithms}}

The concept of the natural logarithm was worked out by [[Gregoire de Saint-Vincent]] and [[Alphonse Antonio de Sarasa]] before 1649.<ref>{{cite book |author-first=R. P. |author-last=Burn |date=2001 |title=Alphonse Antonio de Sarasa and Logarithms |publisher=[[Historia Mathematica]] |pages=28:1–17}}</ref> Their work involved [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] {{nowrap|1=''xy'' = 1}} by determination of the area of [[hyperbolic sector]]s. Their solution generated the requisite "hyperbolic logarithm" [[function (mathematics)|function]] having properties now associated with the natural logarithm.

An early mention of the natural logarithm was by [[Nicholas Mercator]] in his work ''Logarithmotechnia'' published in 1668,<ref>{{cite web |author-first1=J. J. |author-last1=O'Connor |author-first2=E. F. |author-last2=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 |access-date=2009-02-02}}</ref> although the mathematics teacher [[John Speidell]] had already in 1619 compiled a table of what in fact were effectively natural logarithms.<ref name = Cajori>{{cite book |author-last=Cajori |author-first=Florian |author-link=Florian Cajori |title=A History of Mathematics |edition=5th |page=152 |publisher=AMS Bookstore |date=1991 |isbn=0-8218-2102-4 |url=https://books.google.com/?id=mGJRjIC9fZgC}}</ref> It has been said that Speidell's logarithms were to the base ''e'', but this is not entirely true due to complications with the values being expressed as integers.<ref name = Cajori/>{{rp|152}}

==Notational conventions==
The notations {{nowrap|"ln ''x''"}} and {{nowrap|"log<sub>''e''</sub> ''x''"}} both refer unambiguously to the natural logarithm of ''x''.
{{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 group="nb">Including [[C (programming language)|C]], [[C++]], [[SAS System|SAS]], [[MATLAB]], [[Mathematica]], <!--[[Pascal programming language|Pascal]], -->[[Fortran]], and some [[BASIC programming language|BASIC]] dialects</ref> In some other contexts, however, {{nowrap|"log ''x''"}} can be used to denote the [[common logarithm|common (base 10) logarithm]].

Historically, the notations "{{math|l.}}" and  "{{math|l}}" were in use at least since the 1730s,<ref name="Euler_1737">{{cite journal |title=Variae observationes circa series infinitas |author-first=Leonhard |author-last=Euler |author-link=Leonhard Euler |journal=Commentarii academiae scientiarum imperialis Petropolitanae (CASP) |volume=9 |date=1737 |publication-date=1744 |pages=160–188 |id=E72}}</ref><ref name="Euler">{{cite book |author-first=Leonhard |author-last=Euler |author-link=Leonhard Euler |title=Opera Omnia, Series Prima: Opera Mathematica |orig-year= |date=1925 |volume=Quartum Decimum |publisher=[[Teubner]]}}</ref> and until at least the 1840s,<ref name="Cauchy">{{cite book |author-first=Augustin |author-last=Cauchy |author-link=Augustin Cauchy |title=Exercices d'analyse et de physique mathématique |volume=3 |page=380 |url=https://books.google.com/books?hl=fr&id=zQo7AAAAcAAJ |access-date=2015-10-31}}</ref> then "log."<ref name="Legendre_179x">{{cite book |author-first=Adrien-Marie |author-last=Legendre |author-link=Adrien-Marie Legendre |title=Essai sur la théorie des nombres |location=Paris, France |publisher=Duprat, libraire pour les mathématiques, quai des Augustins |volume=VI |date=1798}}</ref> or "log",<ref name="Landau_1909">{{cite book |author-first=Edmund |author-last=Landau |author-link=Edmund Landau |title=Handbuch der Lehre von der Verteilung der Primzahlen |location=Berlin |orig-year=1909 |date=1953 |edition=2 |publisher=Chelsea, New York}}</ref> at least since the 1790s. Finally, in the twentieth century, the notations "Log"<ref name="Piskounov_1972">{{cite book |author-first=Nikolaï |author-last=Piskounov |author-link=:fr:Nikolaï Piskounov |title=Calcul différentiel et intégral |edition=5 |date=1972 |publisher=Editions Mir |location=Moskow |page=91 |url=http://fr.scribd.com/doc/26412181/Calcul-Differentiel-Et-Integral-Tome1-N-Piskounov-Mir}}</ref> and "logh"<ref name="Jolley_1961">{{cite book |author-first=L. B. W. |author-last=Jolley |date=1961 |title=Summation of Series |edition=2 (revised) |publisher=[[Dover Publications, Inc.]] |location=New York, USA |lccn=61-65274 |url=http://plouffe.fr/simon/math/SummationofSeries.pdf |access-date=2015-10-31}}</ref> are attested.

==Definitions==
[[File: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.]]
[[File:Log.gif|The area under the hyperbola satisfies the logarithm rule.  Here ''A''(''s'',''t'') denotes the area under the hyperbola between ''s'' and ''t''.|right|thumb]]

Formally, ln(''a'') may be defined as the area under the [[Hyperbola#Rectangular hyperbola|hyperbola]] 1/''x''.  This is the [[integral]],
:<math>\ln a = \int_1^a \frac{1}{x}\,dx.</math>

This function is a logarithm because it satisfies the fundamental property of a logarithm:
:<math>\ln(ab) = \ln a + \ln b.</math>

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:
:<math>\begin{align}
\ln(ab)= \int_1^{ab}\frac{1}{x} \, dx
&=\int_1^a \frac 1 x \, dx + \int_a^{ab} \frac{1}{x} \, dx\\[5pt]
&=\int_1^a \frac 1 x \, dx + \int_1^b \frac{1}{at} \, d(at)\\[5pt]
&=\int_1^a \frac 1 x \, dx + \int_1^b \frac{1}{t} \, dt\\[5pt]
&= \ln a + \ln b.
\end{align}</math>

In elementary terms, this is simply scaling by 1/''a'' in the horizontal direction and by ''a'' in the vertical direction.  Area does not change under this transformation, but the region between ''a'' and ''ab'' is reconfigured.  Because the function ''a''/(''ax'') is equal to the function 1/''x'', the resulting area is precisely ln(''b'').

The number ''[[E (mathematical constant)|e]]'' is defined as the unique real number ''a'' such that ln(''a'')&nbsp;=&nbsp;1.

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''))&nbsp;=&nbsp;''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&nbsp;''x''.

==Properties==
* <math>\ln 1 = 0</math>
* <math>\ln e = 1</math>
* <math>\ln(xy) = \ln x + \ln y \quad \text{for }\; x > 0\;\text{and }\; y > 0</math>
* <math>\ln(x^y) = y \ln x \quad \text{for }\; x > 0</math>
* <math>\ln x < \ln y \quad\text{for }\; 0 < x < y</math>
* <math>\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1</math>

{{Collapse top|title=Proof|width=80%|left=yes}}
:<math>\lim_{h \to 0} \frac{\ln(1+h)}{h} = \lim_{h \to 0} \frac{\ln(1+h)-\ln 1}{h} = \frac{d}{dx} \ln x \Bigg|_{x=1} = 1</math>
{{Collapse bottom}}

* <math>\lim_{\alpha \to 0} \frac{x^\alpha-1}{\alpha} = \ln x\quad \text{for }\; x > 0</math>

{{Collapse top|title=Proof|width=80%|left=yes}}
Since <math>\ln</math> is the inverse of the exponential function <math>e^x</math>, one can write
:<math> \lim_{\alpha \to 0} \frac{x^\alpha-1}{\alpha}
    = \ln x \cdot \lim_{\alpha \to 0} \frac{e^{\ln x\cdot\alpha}-1}{\ln x\cdot\alpha}
  = \ln x \cdot \lim_{\beta \to 0} \frac{e^{\beta}-1}{\beta}  = \ln x,
</math>
which proves the claim.
{{Collapse bottom}}

* <math>\frac{x-1}{x} \leq \ln x \leq x-1 \quad\text{for}\quad x > 0</math>
* <math>\ln{( 1+x^\alpha )} \leq \alpha x \quad\text{for}\quad x \ge 0\;\text{and }\; \alpha \ge 1</math>

{{Collapse top|title=Proof|width=80%|left=yes}}
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

:<math>\frac{d}{dx} \ln{( 1+x^\alpha )} = \frac{\alpha x^{\alpha - 1}}{1 + x^\alpha} \leq \alpha = \frac{d}{dx} ( \alpha x ) </math>

(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

:<math> x^{\alpha-1} \leq x^\alpha + 1 </math>
:<math> x^{\alpha-1} (1-x) \leq 1 </math>

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.

An alternate proof is to observe that <math>(1+x^\alpha)\leq (1+x)^\alpha </math> under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using <math> \ln(1+x)\leq x </math> completes the proof.
{{Collapse bottom}}

== Derivative ==

The [[derivative]] of the natural logarithm as a real-valued function on the positive reals is given by
:<math>\frac{d}{dx} \ln x = \frac{1}{x}.</math>

How to establish this derivative of the natural logarithm depends on how it is defined firsthand.  If the natural logarithm is defined as the integral
:<math>\ln x = \int_1^x \frac{1}{t}\,dt,</math>
then the derivative immediately follows from the first part of the [[Fundamental theorem of calculus#First part|fundamental theorem of calculus]].

If the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative for ''x'' > 0 can be found by using the properties of the logarithm and a definition of the exponential function. 
:<math>\begin{align}
   \frac{d}{dx} \ln x &= \lim_{h\to 0} \frac{\ln(x+h) - \ln x}{h} \\
   &= \lim_{h\to 0}\left( \frac{1}{h} \ln\left(\frac{x+h}{x}\right)\right) \\
   &= \lim_{h\to 0} \ln\left(1 + \frac{h}{x}\right)^{\frac{1}{h}} \quad &&\text{all above for logarithmic properties}\\
   &= \ln \lim_{h\to 0} \left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\quad &&\text{for continuity of the logarithm} \\
   &= \ln e^{1/x} \quad &&\text{for the definition of } e^x= \lim_{h\to 0}(1 + hx)^{(1/h)}\\
   &= \frac{1}{x} \quad &&\text{for the definition of the ln as inverse function.}
 \end{align}</math>

== Series ==

[[File:LogTay.svg|290px|thumb|right|The Taylor polynomials for ln(1&nbsp;+&nbsp;''x'') only provide accurate approximations in the range −1&nbsp;&lt;&nbsp;''x''&nbsp;≤&nbsp;1. Beyond some ''x''&nbsp;&gt;&nbsp;1, the Taylor polynomials of higher degree are increasingly ''worse'' approximations.]]

If <math>\vert x - 1 \vert \leq 1 \text{ and } x \neq 0, </math> then<ref>[http://www.math2.org/math/expansion/log.htm "Logarithmic Expansions" at Math2.org]</ref>

:<math>\begin{align}
   \ln x &= \int_1^x \frac{1}{t} \, dt = \int_0^{x - 1} \frac{1}{1 + u} \, du \\
   &= \int_0^{x - 1} (1 - u + u^2 - u^3 + \cdots) \, du \\
   &= (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \frac{(x - 1)^4}{4} + \cdots \\
   &= \sum_{k=1}^\infty \frac{(-1)^{k-1} (x-1)^k}{k}.
 \end{align}</math>

This is the [[Taylor series]] for ln&nbsp;''x'' around 1. A change of variables yields the [[Mercator series]]:

:<math>\ln(1+x)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} x^k = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots,</math>

valid for |''x''|&nbsp;&le;&nbsp;1 and ''x''&nbsp;≠&nbsp;&minus;1.

[[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> disregarding <math>x\ne -1</math>, nevertheless applied this series to ''x''&nbsp;=&nbsp;−1, in order to show that the [[harmonic series (mathematics)|harmonic series]] equals the (natural) logarithm of 1/(1&nbsp;−&nbsp;1), that is the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at ''N'' is close to the logarithm of ''N'', when ''N'' is large.

At right is a picture of ln(1&nbsp;+&nbsp;''x'') and some of its [[Taylor polynomial]]s around 0. These approximations converge to the function only in the region −1&nbsp;&lt;&nbsp;''x''&nbsp;≤&nbsp;1; outside of this region the higher-degree Taylor polynomials evolve to ''worse'' approximations for the function.

A useful special case for positive integers ''n'', taking <math>x=\tfrac{1}{n}</math>, is:
:<math> \ln \left(\frac{n + 1}{n}\right) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k n^k}  = \frac{1}{n} - \frac{1}{2 n^2} + \frac{1}{3 n^3} - \frac{1}{4 n^4} + \cdots</math>

If <math>\operatorname{Re}(x) \ge 1/2,</math> then
:<math>\begin{align}
   \ln (x) &= - \ln \left(\frac{1}{x}\right) = - \sum_{k=1}^\infty \frac{(-1)^{k-1} (\frac{1}{x} - 1)^k}{k} = \sum_{k=1}^\infty \frac{(x - 1)^k}{k x^k} \\
   &= \frac{x - 1}{x} + \frac{(x - 1)^2}{2 x^2} +  \frac{(x - 1)^3}{3 x^3} + \frac{(x - 1)^4}{4 x^4} + \cdots
 \end{align}</math>

Now taking  <math>x=\tfrac{n+1}{n}</math> for positive integers ''n'', yields:
:<math> \ln \left(\frac{n + 1}{n}\right) = \sum_{k=1}^\infty \frac{1}{k (n + 1)^k} = \frac{1}{n + 1} + \frac{1}{2 (n + 1)^2} + \frac{1}{3 (n + 1)^3} + \frac{1}{4 (n + 1)^4} + \cdots</math>

If <math>\operatorname{Re}(x) \ge 0 \text{ and } x \neq 0,</math> then
:<math> \ln (x) = \ln \left(\frac{2x}{2}\right) = \ln\left(\frac{1 + \frac{x - 1}{x + 1}}{1 - \frac{x - 1}{x + 1}}\right) = \ln \left(1 + \frac{x - 1}{x + 1}\right) - \ln \left(1 - \frac{x - 1}{x + 1}\right). </math>
Since
:<math>\begin{align}
\ln(1+y) - \ln(1-y)&= \sum^\infty_{i=1}\frac{1}{i}\left((-1)^{i-1}y^i - (-1)^{i-1}(-y)^i\right) = \sum^\infty_{i=1}\frac{y^i}{i}\left((-1)^{i-1} +1\right)  \\
&= y\sum^\infty_{i=1}\frac{y^{i-1}}{i}\left((-1)^{i-1} +1\right)\overset{i-1\to 2k}{=}\; 2y\sum^\infty_{k=0}\frac{y^{2k}}{2k+1},
\end{align}</math> 
we arrive at
:<math>\begin{align}
   \ln (x) &= \frac{2(x - 1)}{x + 1} \sum_{k = 0}^\infty \frac{1}{2k + 1} {\left(\frac{(x - 1)^2}{(x + 1)^2}\right)}^k \\
&= \frac{2(x - 1)}{x + 1} \left( \frac{1}{1} + \frac{1}{3} \frac{(x - 1)^2}{(x + 1)^2} + \frac{1}{5} {\left(\frac{(x - 1)^2}{(x + 1)^2}\right)}^2 + \cdots \right) .
 \end{align}</math>
Substituting again <math>x=\tfrac{n+1}{n}</math> for positive integers ''n'', yields: 
:<math>\begin{align}
   \ln \left(\frac{n + 1}{n}\right) &= \frac{2}{2n + 1} \sum_{k=0}^\infty \frac{1}{(2k + 1) ((2n + 1)^2)^k}\\
&= 2 \left(\frac{1}{2n + 1} + \frac{1}{3 (2n + 1)^3} + \frac{1}{5 (2n + 1)^5} + \cdots \right).
\end{align}</math>

This is, by far, the fastest converging of the series described here.

==The natural logarithm in integration==
The natural logarithm allows simple [[integral|integration]] of functions of the form ''g''(''x'') = ''f''&nbsp;'(''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:

:<math>\frac{d}{dx}\ln \left| x \right| = \frac{1}{x}.</math>

In other words,

:<math>\int \frac{1}{x} \,dx = \ln|x| + C</math>

and

:<math>\int { \frac{f'(x)}{f(x)}\,dx} = \ln|f(x)| + C.</math>

Here is an example in the case of ''g''(''x'') = tan(''x''):

: <math>
\begin{align}
& \int \tan x \,dx = \int \frac{\sin x}{\cos x} \,dx \\[6pt]
& \int \tan x \,dx = \int \frac{-\frac{d}{dx} \cos x}{\cos x} \,dx.
\end{align}
</math>
Letting ''f''(''x'') = cos(''x''):
:<math>\int \tan x \,dx = -\ln \left| \cos x \right| + C</math>
:<math>\int \tan x \,dx = \ln \left| \sec x \right| + C</math>

where ''C'' is an [[arbitrary constant of integration]].

The natural logarithm can be integrated using [[integration by parts]]:

:<math>\int \ln x \,dx = x \ln x - x + C.</math>

Let:

:<math>u = \ln x \Rightarrow du = \frac{dx}{x}</math>
:<math>dv = dx  \Rightarrow v = x</math>

then:

: <math>
\begin{align}
\int \ln x \,dx & = x \ln x - \int \frac{x}{x} \,dx \\
& = x \ln x - \int 1 \,dx \\
& = x \ln x - x + C
\end{align}
</math>

==Numerical value==<!-- This section is linked from [[Common logarithm]] -->

For ln(''x'') where ''x''&nbsp;>&nbsp;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:
:<math>\begin{align}
\ln 123.456 &= \ln(1.23456 \cdot 10^2)\\
&= \ln 1.23456 + \ln(10^2)\\
&= \ln 1.23456 + 2 \ln 10\\
&\approx \ln 1.23456 + 2 \cdot 2.3025851.
\end{align}</math>

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

===Natural logarithm of 10===
The natural logarithm of 10, which has the decimal expansion 2.30258509...,<ref>{{OEIS2C|A002392}}</ref> plays a role for example in the computation of natural logarithms of numbers represented in [[scientific notation]], as a mantissa multiplied by a power of 10:

: <math>\ln(a\cdot 10^n) = \ln a + n \ln 10.</math>

This means that one can effectively calculate the logarithms of numbers with very large or very small [[magnitude (mathematics)|magnitude]] using the logarithms of a relatively small set of decimals in the range <math>[1,10)</math>.

==={{anchor|lnp1}}High precision===
To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if {{math|''x''}} is near 1, a good alternative is to use [[Halley's method]] or [[Newton's method]] to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of {{math|''y''}} to give {{math|1=exp(''y'') − ''x'' = 0}} using Halley's method, or equivalently to give {{math|1=exp(''y''/2) − ''x'' exp(−''y''/2) = 0}} using Newton's method, the iteration simplifies to
:<math> y_{n+1} = y_n + 2 \cdot \frac{ x - \exp ( y_n ) }{ x + \exp ( y_n ) } </math>
which has [[cubic convergence]] to {{math|ln(''x'')}}.

Another alternative for extremely high precision calculation is the formula<ref>{{cite journal |author-first1=T. |author-last1=Sasaki |author-first2=Y. |author-last2=Kanada |title=Practically fast multiple-precision evaluation of log(x) |journal=Journal of Information Processing |volume=5 |issue=4 |pages=247–250 |date=1982 |url=http://ci.nii.ac.jp/naid/110002673332 |access-date=2011-03-30}}</ref>
<ref>{{cite journal |author-first1=Timm |author-last1=Ahrendt |work=Stacs 99 |series=Lecture Notes in Computer Science |doi=10.1007/3-540-49116-3_28 |volume=1564 |date=1999 |pages=302–312 |title=Fast Computations of the Exponential Function |isbn=978-3-540-65691-3}}</ref>

:<math>\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2,</math>

where {{math|''M''}} denotes the [[arithmetic-geometric mean]] of 1 and {{math|4/''s''}}, and

:<math>s = x 2^m > 2^{p/2},</math>

with {{math|''m''}} chosen so that {{math|''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.)

Based on a proposal by [[William Kahan]] and first implemented in the [[Hewlett-Packard]] [[HP-41C]] calculator in 1979 (referred to under "LN1" in the display, only), some calculators, [[operating system]]s (for example [[Berkeley UNIX 4.3BSD]]<ref name="Beebe_2017"/>), [[computer algebra system]]s and programming languages (for example [[C99]]<ref name="Beebe_2002"/>) provide a special '''natural logarithm plus 1''' function, alternatively named '''LNP1''',<ref name="HP48_AUR">{{cite book |title=HP&nbsp;48G Series – Advanced User's Reference Manual (AUR) |publisher=[[Hewlett-Packard]] |edition=4 |date=December 1994 |id=HP 00048-90136, 0-88698-01574-2 |orig-year=1993<!-- edition 1 (1993-07) --> |url=http://www.hpcalc.org/details.php?id=6036 |access-date=2015-09-06}}</ref><ref name="HP50_AUR">{{cite book |title=HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) |publisher=[[Hewlett-Packard]] |edition=2 |date=2009-07-14 |orig-year=2005<!-- first published: Edition 1 (2005–09) --> |id=HP F2228-90010 |url=http://www.hpcalc.org/details.php?id=7141 | accessdate=2015-10-10}} [http://holyjoe.net/hp/HP_50g_AUR_v2_English_searchable.pdf Searchable PDF]</ref> or '''log1p'''<ref name="Beebe_2002"/> to give more accurate results for logarithms close to zero by passing arguments ''x'', also close to zero, to a function log1p(''x''), which returns the value ln(1+''x''), instead of passing a value ''y'' close to 1 to a function returning ln(''y'').<ref name="Beebe_2002">{{cite web |title=Computation of expm1 = exp(x)−1 |author-first=Nelson H. F. |author-last=Beebe |publisher=Department of Mathematics, Center for Scientific Computing, [[University of Utah]] |location=Salt Lake City, Utah, USA |date=2002-07-09 |version=1.00 |url=http://www.math.utah.edu/~beebe/reports/expm1.pdf |access-date=2015-11-02}}</ref><ref name="HP48_AUR"/><ref name="HP50_AUR"/> The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln, thereby allowing for a high accuracy for both the argument and the result near zero.<ref name="HP48_AUR"/><ref name="HP50_AUR"/>

In addition to base {{math|''e''}} the [[IEEE 754-2008]] standard defines similar logarithmic functions near 1 for [[binary logarithm|binary]] and [[decimal logarithm]]s: <math>\log_{2} (1 + x)</math> and <math>\log_{10} (1 + x)</math>.

Similar inverse functions named "[[expm1]]",<ref name="Beebe_2002"/> "expm"<ref name="HP48_AUR"/><ref name="HP50_AUR"/> or "exp1m" exist as well, all with the meaning of {{nowrap|expm1(''x'') {{=}} exp(''x'') - 1.}}<ref group="nb" name="Alternative_funcs">For a similar approach to reduce [[round-off error]]s of calculations for certain input values see [[trigonometric function]]s like [[versine]], [[vercosine]], [[coversine]], [[covercosine]], [[haversine]], [[havercosine]], [[hacoversine]], [[hacovercosine]], [[exsecant]] and [[excosecant]].</ref>

An identity in terms of the [[artanh|inverse hyperbolic tangent]],
:<math>\mathrm{log1p}(x) = \log(1+x) = 2 ~ \mathrm{artanh}\left(\frac{x}{2+x}\right)\,,</math>
gives a high precision value for small values of {{math|''x''}} on systems that do not implement {{math|log1p(''x'')}}.

===Computational complexity===
{{main|Computational complexity of mathematical operations}}
The [[Computational complexity theory|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.

==Continued fractions==
While no simple [[continued fraction]]s are available, several [[generalized continued fraction]]s are, including:

:<math>
\begin{align}
\ln(1+x) & =\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots \\[5pt]
& = \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}}}}}
\end{align}
</math>

: <math>
\begin{align}
\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}}}}}} \\[5pt]
& = \cfrac{2x} {2y+x-\cfrac{(1x)^2} {3(2y+x)-\cfrac{(2x)^2} {5(2y+x)-\cfrac{(3x)^2} {7(2y+x)-\ddots}}}}
\end{align}
</math>

These continued fractions—particularly the last—converge rapidly for values close to&nbsp;1. However, the natural logarithms of much larger numbers can easily be computed by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.25<sup>3</sup> × 1.024, the [[natural logarithm of 2]] can be computed as:

: <math>
\begin{align}
\ln 2 & = 3 \ln\left(1+\frac{1}{4}\right) + \ln\left(1+\frac{3}{125}\right) \\[8pt]
& = \cfrac{6} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{6} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
\end{align}
</math>

Furthermore, since 10 = 1.25<sup>10</sup> × 1.024<sup>3</sup>, even the natural logarithm of 10 similarly can be computed as:

: <math>
\begin{align}
\ln 10 & = 10 \ln\left(1+\frac{1}{4}\right) + 3\ln\left(1+\frac{3}{125}\right) \\[10pt]
& = \cfrac{20} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{18} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
\end{align}
</math>

==Complex logarithms==
{{Main|Complex logarithm}}
The exponential function can be extended to a function which gives a [[complex number]] as {{math|''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 {{math|''e''<sup>''x''</sup> {{=}} 0}}; and it turns out that {{math|''e''<sup>2{{pi}}''i''</sup> {{=}} 1 {{=}} ''e''<sup>0</sup>}}. Since the multiplicative property still works for the complex exponential function, {{math|''e''<sup>''z''</sup> {{=}} ''e''<sup>''z''+2{{pi}}''ki''</sup>}}, for all complex ''z'' and integers&nbsp;''k''.

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{{pi}}''i'' at will. The complex logarithm can only be single-valued on the [[complex plane#Cutting the plane|cut plane]].  For example, {{math|ln(''i'')}} = {{sfrac|{{pi}}''i''|2}} or {{sfrac|5{{pi}}''i''|2}} or -{{sfrac|3{{pi}}''i''|2}}, etc.; and although {{math|''i''<sup>4</sup> {{=}} 1, 4 log(''i'')}} can be defined as 2{{pi}}''i'', or 10{{pi}}''i'' or −6{{pi}}''i'', and so on.

<gallery mode="packed" caption="Plots of the natural logarithm function on the complex plane ([[principal branch]])">
Image:NaturalLogarithmRe.png| ''z'' = Re(ln(''x'' + ''yi''))
Image:NaturalLogarithmImAbs.png| ''z'' = abs(Im(ln(''x'' + ''yi'')))
Image:NaturalLogarithmAbs.png| ''z'' = abs(ln(''x'' + ''yi''))
Image:NaturalLogarithmAll.png| Superposition of the previous three graphs
</gallery>

==See also==
* [[Mental calculation#nat-exp|Approximating natural exponents (log base e)]]
* [[Napierian logarithm]]
* [[Logarithm of a matrix]]
* [[Logarithmic integral function]]
* [[Nicholas Mercator]] – first to use the term natural logarithm
* [[Polylogarithm]]
* [[Von Mangoldt function]]
* [[e (mathematical constant)|The number ''e'']]

==Notes==
{{Reflist|group="nb"}}

==References==
{{Reflist|refs=
<ref name="Beebe_2017">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter 10.4. Logarithm near one |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=[[Springer International Publishing AG]] |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |pages=290–292 |quote=In 1987, Berkeley UNIX 4.3BSD introduced the log1p() function}}</ref>
}}

{{DEFAULTSORT:Natural Logarithm}}
[[Category:Logarithms]]
[[Category:Elementary special functions]]
[[Category:E (mathematical constant)]]
[[Category:Unary operations]]

[[de:Logarithmus#Natürlicher Logarithmus]]