Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant [[e (mathematical constant)|Template:Mvar]], which is an irrational and transcendental number approximately equal to Template:Math.<ref>Template:Cite OEIS</ref> The natural logarithm of Template:Mvar is generally written as Template:Math, Template:Math, or sometimes, if the base Template:Mvar is implicit, simply Template:Math.<ref>G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest".</ref><ref>Template:Cite book Extract of page 9</ref> Parentheses are sometimes added for clarity, giving Template:Math, Template:Math, or Template:Math. This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of Template:Mvar is the power to which Template:Mvar would have to be raised to equal Template:Mvar. For example, Template:Math is Template:Math, because Template:Math. The natural logarithm of Template:Mvar itself, Template:Math, is Template:Math, because Template:Math, while the natural logarithm of Template:Math is Template:Math, since Template:Math.

The natural logarithm can be defined for any positive real number Template:Mvar as the area under the curve Template:Math from Template:Math to Template:Mvar<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> (with the area being negative when Template:Math). 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 then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more.

The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: <math display="block">\begin{align}

  e^{\ln x} &= x \qquad \text{ if } x \in \R_{+}\\
  \ln e^x   &= x \qquad \text{ if } x \in \R
\end{align}</math>

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:<ref name=":2">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block"> \ln( x \cdot y ) = \ln x + \ln y~.</math>

Logarithms can be defined for any positive base other than 1, not only Template:Mvar. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, <math>\log_b x = \ln x / \ln b = \ln x \cdot \log_b e</math>.

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 scientific disciplines, and are used to solve problems involving compound interest.

History

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649.<ref>Template:Cite journal</ref> Their work involved quadrature of the hyperbola with equation Template:Math, by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the 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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.<ref name = Cajori>Template:Cite book</ref> It has been said that Speidell's logarithms were to the base Template:Mvar, but this is not entirely true due to complications with the values being expressed as integers.<ref name = Cajori/>Template:Rp

Notational conventions

The notations Template:Math and Template:Math both refer unambiguously to the natural logarithm of Template:Mvar, and Template:Math without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages.<ref group="nb"> Including C, C++, SAS, MATLAB, Mathematica, Fortran, and some BASIC dialects</ref> In some other contexts such as chemistry, however, Template:Math can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.

Generally, the notation for the logarithm to base Template:Math of a number Template:Math is shown as Template:Math. So the Template:Math of Template:Math to the base Template:Math would be Template:Math.

Definitions

The natural logarithm can be defined in several equivalent ways.

Inverse of exponential

The most general definition is as the inverse function of <math>e^x</math>, so that <math>e^{\ln(x)} = x</math>. Because <math>e^x</math> is positive and invertible for any real input <math>x</math>, this definition of <math>\ln(x)</math> is well defined for any positive Template:Mvar.

Integral definition

File:Log-pole-x 1.svg

Template:Math as the area of the shaded region under the curve Template:Math from Template:Math to Template:Mvar. If Template:Mvar is less than Template:Math, the area taken to be negative.

File:Log.gif

The area under the hyperbola satisfies the logarithm rule. Here Template:Math denotes the area under the hyperbola between Template:Mvar and Template:Mvar.

The natural logarithm of a positive, real number Template:Mvar may be defined as the area under the graph of the hyperbola with equation Template:Math between Template:Math and Template:Math. This is the integral<ref name=":1" /> <math display="block">\ln a = \int_1^a \frac{1}{x}\,dx.</math> If Template:Mvar is in <math>(0,1)</math>, then the region has negative area, and the logarithm is negative.

This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:<ref name=":2" /> <math display="block">\ln(ab) = \ln a + \ln b.</math>

This can be demonstrated by splitting the integral that defines Template:Math into two parts, and then making the variable substitution Template:Math (so Template:Math) in the second part, as follows: <math display="block">\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} a\,dt\\[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 Template:Math in the horizontal direction and by Template:Mvar in the vertical direction. Area does not change under this transformation, but the region between Template:Mvar and Template:Math is reconfigured. Because the function Template:Math is equal to the function Template:Math, the resulting area is precisely Template:Math.

The number Template:Mvar can then be defined to be the unique real number Template:Mvar such that Template:Math.

Properties

The natural logarithm has the following mathematical 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) = \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(\sqrt[y]{x}) = (\ln x) / y\quad \text{for }\; x > 0\;\text{and }\; y \ne 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>
• <math>\lim_{\alpha \to 0} \frac{x^\alpha-1}{\alpha} = \ln x\quad \text{for }\; x > 0</math>
• <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>

Derivative

The derivative of the natural logarithm as a real-valued function on the positive reals is given by<ref name=":1" /> <math display="block">\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 display="block">\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.

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for Template:Math) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the number <math>e = \lim_{u\to 0}(1+u)^{1/u},</math> the exponential function can be defined as <math display="block">e^x = \lim_{u\to 0} (1+u)^{x/u} = \lim_{h\to 0}(1 + hx)^{1/h} , </math> where <math>u=hx, h=\frac{u}{x}.</math>

The derivative can then be found from first principles. <math display="block">\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}\left[ \ln\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\quad &&\text{all above for logarithmic properties}\\
  &= \ln \left[ \lim_{h\to 0}\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\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>

Also, we have: <math display="block">\frac{d}{dx} \ln ax = \frac{d}{dx} (\ln a + \ln x) = \frac{d}{dx} \ln a +\frac{d}{dx} \ln x = \frac{1}{x}.</math>

so, unlike its inverse function <math>e^{ax}</math>, a constant in the function doesn't alter the differential.

Series

File:LogTay.svg

The Taylor polynomials for Template:Math only provide accurate approximations in the range Template:Math. Beyond some Template:Math, the Taylor polynomials of higher degree are increasingly worse approximations.

Since the natural logarithm is undefined at 0, <math>\ln(x)</math> itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if <math>\vert x - 1 \vert \leq 1 \text{ and } x \neq 0, </math> then<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\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 <math>\ln x</math> around 1. A change of variables yields the Mercator series: <math display="block">\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 <math>|x| \leq 1</math> and <math>x\ne -1.</math>

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 <math>x=-1</math> to show that the harmonic series equals the natural logarithm of <math>\frac{1}{1-1}</math>; that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at Template:Mvar is close to the logarithm of Template:Mvar, when Template:Mvar is large, with the difference converging to the Euler–Mascheroni constant.

The figure is a graph of Template:Math and some of its Taylor polynomials around 0. These approximations converge to the function only in the region Template:Math; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function.

A useful special case for positive integers Template:Mvar, taking <math>x = \tfrac{1}{n}</math>, is: <math display="block"> \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 display="block">\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 Template:Mvar, we get: <math display="block"> \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 display="block"> \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 display="block">\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 display="block">\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>

Using the substitution <math>x=\tfrac{n+1}{n}</math> again for positive integers Template:Mvar, we get: <math display="block">\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 can also be expressed as an infinite product:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }} (Page 3654, equation 2.6)</ref> <math display="block">\ln(x)=(x-1) \prod_{k=1}^\infty \left ( \frac{2}{1+\sqrt[2^k]{x}} \right )</math>

Two examples might be: <math display="block">\ln(2)=\left ( \frac{2}{1+\sqrt{2}} \right )\left ( \frac{2}{1+\sqrt[4]{2}} \right )\left ( \frac{2}{1+\sqrt[8]{2}} \right )\left ( \frac{2}{1+\sqrt[16]{2}} \right )...</math> <math display="block">\pi=(2i+2)\left ( \frac{2}{1+\sqrt{i}} \right )\left ( \frac{2}{1+\sqrt[4]{i}} \right )\left ( \frac{2}{1+\sqrt[8]{i}} \right )\left ( \frac{2}{1+\sqrt[16]{i}} \right )...</math>

From this identity, we can easily get that: <math display="block">\frac{1}{\ln(x)}=\frac{x}{x-1}-\sum_{k=1}^\infty\frac{2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}</math>

For example: <math display="block">\frac{1}{\ln(2)} = 2-\frac{\sqrt{2}}{2+2\sqrt{2}}-\frac{\sqrt[4]{2}}{4+4\sqrt[4]{2}}-\frac{\sqrt[8]{2}}{8+8\sqrt[8]{2}} \cdots</math>

The natural logarithm in integration

The natural logarithm allows simple integration of functions of the form <math>g(x) = \frac{f'(x)}{f(x)}</math>: an antiderivative of Template:Math is given by <math>\ln (|f(x)|)</math>. This is the case because of the chain rule and the following fact: <math display="block">\frac{d}{dx}\ln \left| x \right| = \frac{1}{x}, \ \ x \ne 0</math>

In other words, when integrating over an interval of the real line that does not include <math>x=0</math>, then <math display="block">\int \frac{1}{x} \,dx = \ln|x| + C</math> where Template:Mvar is an arbitrary constant of integration.<ref>For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, Calculus and Analytic Geometry, 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.</ref>

Likewise, when the integral is over an interval where <math>f(x) \ne 0</math>,

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

For example, consider the integral of <math>\tan (x)</math> over an interval that does not include points where <math>\tan (x)</math> is infinite: <math display="block">\int \tan x \,dx = \int \frac{\sin x}{\cos x} \,dx = -\int \frac{\frac{d}{dx} \cos x}{\cos x} \,dx = -\ln \left| \cos x \right| + C = \ln \left| \sec x \right| + C. </math>

The natural logarithm can be integrated using integration by parts: <math display="block">\int \ln x \,dx = x \ln x - x + C.</math>

Let: <math display="block">u = \ln x \Rightarrow du = \frac{dx}{x}</math> <math display="block">dv = dx \Rightarrow v = x</math> then: <math display="block"> \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>

Efficient computation

For <math>\ln (x)</math> where Template:Math, the closer the value of Template:Mvar is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this: <math display="block">\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, approximately equal to Template:Math,<ref>Template:Cite OEIS</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 display="block">\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 using the logarithms of a relatively small set of decimals in the range Template:Math.

Template:AnchorHigh 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 Template:Mvar 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 Template:Mvar to give <math>\exp(y)-x=0</math> using Halley's method, or equivalently to give <math>\exp(y/2) -x \exp(-y/2)=0</math> using Newton's method, the iteration simplifies to <math display="block"> 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)</math>.

Another alternative for extremely high precision calculation is the formula<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> <math display="block">\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2,</math> where Template:Mvar denotes the arithmetic-geometric mean of 1 and Template:Math, and <math display="block">s = x 2^m > 2^{p/2},</math> with Template:Mvar chosen so that Template:Mvar bits of precision is attained. (For most purposes, the value of 8 for Template:Mvar 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 <math>\ln 2</math> and [[pi|Template:Pi]] can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used: <math display="block">\ln x = \frac{\pi}{M\left(\theta_2^2(1/x),\theta_3^2(1/x)\right)},\quad x\in (1,\infty)</math>

where <math display="block"> \theta_2(x) = \sum_{n\in\Z} x^{(n+1/2)^2},

\quad

\theta_3(x) = \sum_{n\in\Z} x^{n^2} </math> are the Jacobi theta functions.<ref>Template:Cite book page 225</ref>

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 systems (for example Berkeley UNIX 4.3BSD<ref name="Beebe_2017"/>), computer algebra systems 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">Template:Cite book</ref><ref name="HP50_AUR">Template:Cite book Searchable PDF</ref> or log1p<ref name="Beebe_2002"/> to give more accurate results for logarithms close to zero by passing arguments Template:Mvar, also close to zero, to a function Template:Math, which returns the value Template:Math, instead of passing a value Template:Mvar close to 1 to a function returning Template:Math.<ref name="Beebe_2002">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="HP48_AUR"/><ref name="HP50_AUR"/> The function Template:Math avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the natural logarithm. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.<ref name="HP48_AUR"/><ref name="HP50_AUR"/>

In addition to base Template:Mvar, the IEEE 754-2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms: Template:Math and Template: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 Template:Math.<ref group="nb" name="Alternative_funcs">For a similar approach to reduce round-off errors of calculations for certain input values see trigonometric functions like versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.</ref>

An identity in terms of the inverse hyperbolic tangent, <math display="block">\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 Template:Mvar on systems that do not implement Template:Math.

Computational complexity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is <math>\text{O}\bigl(M(n) \ln n \bigr)</math>. Here, Template:Mvar is the number of digits of precision at which the natural logarithm is to be evaluated, and Template:Math is the computational complexity of multiplying two Template:Mvar-digit numbers.

Continued fractions

While no simple continued fractions are available, several generalized continued fractions exist, including: <math display="block"> \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 display="block"> \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 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 = 1.25³ × 1.024, the natural logarithm of 2 can be computed as: <math display="block"> \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 = 1.25¹⁰ × 1.024³, even the natural logarithm of 10 can be computed similarly as: <math display="block"> \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> The reciprocal of the natural logarithm can be also written in this way: <math display="block">\frac {1}{\ln(x)} = \frac {2x}{x^2-1}\sqrt{\frac {1}{2}+\frac {x^2+1}{4x}}\sqrt{\frac {1}{2}+\frac {1}{2}\sqrt{\frac {1}{2}+\frac {x^2+1}{4x}}}\ldots</math>

For example: <math display="block">\frac {1}{\ln(2)} = \frac {4}{3}\sqrt{\frac {1}{2} + \frac {5}{8}} \sqrt{\frac {1}{2} + \frac {1}{2} \sqrt{\frac {1}{2} +\frac {5}{8}}} \ldots</math>

Complex logarithms

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The exponential function can be extended to a function which gives a complex number as Template:Math for any arbitrary complex number Template:Mvar; simply use the infinite series with Template:Mvar=z 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 Template:Mvar has Template:Math; and it turns out that Template:Math. Since the multiplicative property still works for the complex exponential function, Template:Math, for all complex Template:Mvar and integers Template:Mvar.

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 Template:Math at will. The complex logarithm can only be single-valued on the cut plane. For example, Template:Math or Template:Math or Template:Math, etc.; and although Template:Math can be defined as Template:Math, or Template:Math or Template:Math, and so on.

• Plots of the natural logarithm function on the complex plane (principal branch)
• NaturalLogarithmRe.png
• NaturalLogarithmImAbs.png
• NaturalLogarithmAbs.png
• NaturalLogarithmAll.png

  Superposition of the previous three graphs

See also

• Iterated logarithm
• Napierian logarithm
• List of logarithmic identities
• Logarithm of a matrix
• Logarithmic coordinates of an element of a Lie group.
• Logarithmic differentiation
• Logarithmic integral function
• Nicholas Mercator – first to use the term natural logarithm
• Polylogarithm
• Von Mangoldt function