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Natural logarithm

From Wikipedia, the free encyclopedia
Logarithm to the base of the mathematical constant e
"Base e" redirects here. For the numbering system which uses "e" as its base, seeNon-integer base of numeration § Base e.

Natural logarithm
Graph of part of the natural logarithm function.
Graph of part of the natural logarithm function. The function slowly grows to positive infinity asx increases, and slowly goes to negative infinity asx approaches 0 ("slowly" as compared to anypower law ofx).
General information
General definitionln⁡x=loge⁡x{\displaystyle \ln x=\log _{e}x}
Motivation of inventionhyperbola quadrature
Fields of applicationPure and applied mathematics
Domain, codomain and image
DomainR>0{\displaystyle \mathbb {R} _{>0}}
CodomainR{\displaystyle \mathbb {R} }
ImageR{\displaystyle \mathbb {R} }
Specific values
Value at +∞+∞
Value at e1
Value at 10
Value at 0−∞
Specific features
Asymptotex=0{\displaystyle x=0}
Root1
Inverseexp⁡x{\displaystyle \exp x}
Derivativeddxln⁡x=1x,x>0{\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0}
Antiderivative∫ln⁡xdx=x(ln⁡x−1)+C{\displaystyle \int \ln x\,dx=x\left(\ln x-1\right)+C}
Part ofa series of articles on the
mathematical constante
Properties
Applications
Defininge
People
Related topics

Thenatural logarithm of a number is itslogarithm to thebase of themathematical constante, which is anirrational andtranscendental number approximately equal to2.718281828459.[1] The natural logarithm ofx is generally written aslnx,logex, or sometimes, if the basee is implicit, simplylogx.[2][3]Parentheses are sometimes added for clarity, givingln(x),loge(x), orlog(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm ofx is thepower to whiche would have to be raised to equalx. For example,ln 7.5 is2.0149..., becausee2.0149... = 7.5. The natural logarithm ofe itself,lne, is1, becausee1 =e, while the natural logarithm of1 is0, sincee0 = 1.

The natural logarithm can be defined for any positivereal numbera as thearea under the curvey = 1/x from1 toa[4] (with the area being negative when0 <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 then be extended to give logarithm values for negative numbers and for all non-zerocomplex numbers, although this leads to amulti-valued function: seecomplex logarithm for more.

The natural logarithm function, if considered as areal-valued function of a positive real variable, is theinverse function of theexponential function, leading to the identities:eln⁡x=x if x∈R+ln⁡ex=x if x∈R{\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \end{aligned}}}

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:[5]ln⁡(x⋅y)=ln⁡x+ln⁡y .{\displaystyle \ln(x\cdot y)=\ln x+\ln y~.}

Logarithms can be defined for any positive base other than 1, not onlye. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter,logb⁡x=ln⁡x/ln⁡b=ln⁡x⋅logb⁡e{\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e}.

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 thehalf-life, decay constant, or unknown time inexponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involvingcompound interest.

History

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Main article:History of logarithms

The concept of the natural logarithm was worked out byGregoire de Saint-Vincent andAlphonse Antonio de Sarasa before 1649.[6] Their work involvedquadrature of thehyperbola with equationxy = 1, by determination of the area ofhyperbolic 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 byNicholas Mercator in his workLogarithmotechnia, published in 1668,[7] although the mathematics teacherJohn Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.[8] It has been said that Speidell's logarithms were to the basee, but this is not entirely true due to complications with the values being expressed asintegers.[8]: 152 

Notational conventions

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The notationslnx andlogex both refer unambiguously to the natural logarithm ofx, andlogx 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 manyprogramming languages.[nb 1] In some other contexts such aschemistry, however,logx can be used to denote thecommon (base 10) logarithm. It may also refer to thebinary (base 2) logarithm in the context ofcomputer science, particularly in the context oftime complexity.

Generally, the notation for the logarithm to baseb of a numberx is shown aslogbx. So thelog of8 to the base2 would belog2 8 = 3.

Definitions

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The natural logarithm can be defined in several equivalent ways.

Inverse of exponential

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The most general definition is as the inverse function ofex{\displaystyle e^{x}}, so thateln⁡(x)=x{\displaystyle e^{\ln(x)}=x}. Becauseex{\displaystyle e^{x}} is positive and invertible for any real inputx{\displaystyle x}, this definition ofln⁡(x){\displaystyle \ln(x)} is well defined for any positivex.

Integral definition

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lna as the area of the shaded region under the curvef(x) = 1/x from1 toa. Ifa is less than1, the area taken to be negative.
The area under the hyperbola satisfies the logarithm rule. HereA(s,t) denotes the area under the hyperbola betweens andt.

The natural logarithm of a positive, real numbera may be defined as thearea under the graph of thehyperbola with equationy = 1/x betweenx = 1 andx =a. This is theintegral[4]ln⁡a=∫1a1xdx.{\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}Ifa is in(0,1){\displaystyle (0,1)}, then the region hasnegative area, and the logarithm is negative.

This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:[5]ln⁡(ab)=ln⁡a+ln⁡b.{\displaystyle \ln(ab)=\ln a+\ln b.}

This can be demonstrated by splitting the integral that defineslnab into two parts, and then making thevariable substitutionx =at (sodx =adt) in the second part, as follows:ln⁡ab=∫1ab1xdx=∫1a1xdx+∫aab1xdx=∫1a1xdx+∫1b1atadt=∫1a1xdx+∫1b1tdt=ln⁡a+ln⁡b.{\displaystyle {\begin{aligned}\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{aligned}}}

In elementary terms, this is simply scaling by1/a in the horizontal direction and bya in the vertical direction. Area does not change under this transformation, but the region betweena andab is reconfigured. Because the functiona/(ax) is equal to the function1/x, the resulting area is preciselylnb.

The numbere can then be defined to be the unique real numbera such thatlna = 1.

Properties

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The natural logarithm has the following mathematical properties:

Derivative

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Thederivative of the natural logarithm as areal-valued function on the positive reals is given by[4]ddxln⁡x=1x.{\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}.}

How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integralln⁡x=∫1x1tdt,{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,}then the derivative immediately follows from the first part of thefundamental theorem of calculus.

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (forx > 0) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the numbere=limu→0(1+u)1/u,{\displaystyle e=\lim _{u\to 0}(1+u)^{1/u},} the exponential function can be defined asex=limu→0(1+u)x/u=limh→0(1+hx)1/h,{\displaystyle e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},} whereu=hx,h=ux.{\displaystyle u=hx,h={\frac {u}{x}}.}

The derivative can then be found from first principles.ddxln⁡x=limh→0ln⁡(x+h)−ln⁡xh=limh→0[1hln⁡(x+hx)]=limh→0[ln⁡(1+hx)1h]all above for logarithmic properties=ln⁡[limh→0(1+hx)1h]for continuity of the logarithm=ln⁡e1/xfor the definition of ex=limh→0(1+hx)1/h=1xfor the definition of the ln as inverse function.{\displaystyle {\begin{aligned}{\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{aligned}}}

Also, we have:ddxln⁡ax=ddx(ln⁡a+ln⁡x)=ddxln⁡a+ddxln⁡x=1x.{\displaystyle {\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}}.}

so, unlike its inverse functioneax{\displaystyle e^{ax}}, a constant in the function doesn't alter the differential.

Series

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The Taylor polynomials forln(1 +x) only provide accurate approximations in the range−1 <x ≤ 1. Beyond somex > 1, the Taylor polynomials of higher degree are increasinglyworse approximations.

Since the natural logarithm is undefined at 0,ln⁡(x){\displaystyle \ln(x)} itself does not have aMaclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if|x−1|≤1 and x≠0,{\displaystyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0,} then[9]ln⁡x=∫1x1tdt=∫0x−111+udu=∫0x−1(1−u+u2−u3+⋯)du=(x−1)−(x−1)22+(x−1)33−(x−1)44+⋯=∑k=1∞(−1)k−1(x−1)kk.{\displaystyle {\begin{aligned}\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{aligned}}}

This is theTaylor series forln⁡x{\displaystyle \ln x} around 1. A change of variables yields theMercator series:ln⁡(1+x)=∑k=1∞(−1)k−1kxk=x−x22+x33−⋯,{\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}valid for|x|≤1{\displaystyle |x|\leq 1} andx≠−1.{\displaystyle x\neq -1.}

Leonhard Euler,[10] disregardingx≠−1{\displaystyle x\neq -1}, nevertheless applied this series tox=−1{\displaystyle x=-1} to show that theharmonic series equals the natural logarithm of11−1{\displaystyle {\frac {1}{1-1}}}; that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated atN is close to the logarithm ofN, whenN is large, with the difference converging to theEuler–Mascheroni constant.

The figure is agraph ofln(1 +x) and some of itsTaylor polynomials around 0. These approximations converge to the function only in the region−1 <x ≤ 1; outside this region, the higher-degree Taylor polynomials devolve toworse approximations for the function.

A useful special case for positive integersn, takingx=1n{\displaystyle x={\tfrac {1}{n}}}, is:ln⁡(n+1n)=∑k=1∞(−1)k−1knk=1n−12n2+13n3−14n4+⋯{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots }

IfRe⁡(x)≥1/2,{\displaystyle \operatorname {Re} (x)\geq 1/2,} thenln⁡(x)=−ln⁡(1x)=−∑k=1∞(−1)k−1(1x−1)kk=∑k=1∞(x−1)kkxk=x−1x+(x−1)22x2+(x−1)33x3+(x−1)44x4+⋯{\displaystyle {\begin{aligned}\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}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \end{aligned}}}

Now, takingx=n+1n{\displaystyle x={\tfrac {n+1}{n}}} for positive integersn, we get:ln⁡(n+1n)=∑k=1∞1k(n+1)k=1n+1+12(n+1)2+13(n+1)3+14(n+1)4+⋯{\displaystyle \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 }

IfRe⁡(x)≥0 and x≠0,{\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,} thenln⁡(x)=ln⁡(2x2)=ln⁡(1+x−1x+11−x−1x+1)=ln⁡(1+x−1x+1)−ln⁡(1−x−1x+1).{\displaystyle \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).}Sinceln⁡(1+y)−ln⁡(1−y)=∑i=1∞1i((−1)i−1yi−(−1)i−1(−y)i)=∑i=1∞yii((−1)i−1+1)=y∑i=1∞yi−1i((−1)i−1+1)=i−1→2k2y∑k=0∞y2k2k+1,{\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\end{aligned}}} we arrive atln⁡(x)=2(x−1)x+1∑k=0∞12k+1((x−1)2(x+1)2)k=2(x−1)x+1(11+13(x−1)2(x+1)2+15((x−1)2(x+1)2)2+⋯).{\displaystyle {\begin{aligned}\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{aligned}}}Using the substitutionx=n+1n{\displaystyle x={\tfrac {n+1}{n}}} again for positive integersn, we get:ln⁡(n+1n)=22n+1∑k=0∞1(2k+1)((2n+1)2)k=2(12n+1+13(2n+1)3+15(2n+1)5+⋯).{\displaystyle {\begin{aligned}\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{aligned}}}

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

The natural logarithm can also be expressed as an infinite product:[11]ln⁡(x)=(x−1)∏k=1∞(21+x2k){\displaystyle \ln(x)=(x-1)\prod _{k=1}^{\infty }\left({\frac {2}{1+{\sqrt[{2^{k}}]{x}}}}\right)}

Two examples might be:ln⁡(2)=(21+2)(21+24)(21+28)(21+216)...{\displaystyle \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)...}π=(2i+2)(21+i)(21+i4)(21+i8)(21+i16)...{\displaystyle \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)...}

From this identity, we can easily get that:1ln⁡(x)=xx−1−∑k=1∞2−kx2−k1+x2−k{\displaystyle {\frac {1}{\ln(x)}}={\frac {x}{x-1}}-\sum _{k=1}^{\infty }{\frac {2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}}}

For example:1ln⁡(2)=2−22+22−244+424−288+828⋯{\displaystyle {\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 }

The natural logarithm in integration

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The natural logarithm allows simpleintegration of functions of the formg(x)=f′(x)f(x){\displaystyle g(x)={\frac {f'(x)}{f(x)}}}: anantiderivative ofg(x) is given byln⁡(|f(x)|){\displaystyle \ln(|f(x)|)}. This is the case because of thechain rule and the following fact:ddxln⁡|x|=1x,  x≠0{\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}},\ \ x\neq 0}

In other words, when integrating over an interval of the real line that does not includex=0{\displaystyle x=0}, then∫1xdx=ln⁡|x|+C{\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C}whereC is anarbitrary constant of integration.[12]

Likewise, when the integral is over an interval wheref(x)≠0{\displaystyle f(x)\neq 0},

∫f′(x)f(x)dx=ln⁡|f(x)|+C.{\displaystyle \int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.}

For example, consider the integral oftan⁡(x){\displaystyle \tan(x)} over an interval that does not include points wheretan⁡(x){\displaystyle \tan(x)} is infinite:∫tan⁡xdx=∫sin⁡xcos⁡xdx=−∫ddxcos⁡xcos⁡xdx=−ln⁡|cos⁡x|+C=ln⁡|sec⁡x|+C.{\displaystyle \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.}

The natural logarithm can be integrated usingintegration by parts:∫ln⁡xdx=xln⁡x−x+C.{\displaystyle \int \ln x\,dx=x\ln x-x+C.}

Let:u=ln⁡x⇒du=dxx{\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}}dv=dx⇒v=x{\displaystyle dv=dx\Rightarrow v=x}then:∫ln⁡xdx=xln⁡x−∫xxdx=xln⁡x−∫1dx=xln⁡x−x+C{\displaystyle {\begin{aligned}\int \ln x\,dx&=x\ln x-\int {\frac {x}{x}}\,dx\\&=x\ln x-\int 1\,dx\\&=x\ln x-x+C\end{aligned}}}

Efficient computation

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Forln⁡(x){\displaystyle \ln(x)} wherex > 1, the closer the value ofx 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:ln⁡123.456=ln⁡(1.23456⋅102)=ln⁡1.23456+ln⁡(102)=ln⁡1.23456+2ln⁡10≈ln⁡1.23456+2⋅2.3025851.{\displaystyle {\begin{aligned}\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{aligned}}}

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

Natural logarithm of 10

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The natural logarithm of 10, approximately equal to2.30258509,[13] plays a role for example in the computation of natural logarithms of numbers represented inscientific notation, as amantissa multiplied by a power of 10:ln⁡(a⋅10n)=ln⁡a+nln⁡10.{\displaystyle \ln(a\cdot 10^{n})=\ln a+n\ln 10.}

This means that one can effectively calculate the logarithms of numbers with very large or very smallmagnitude using the logarithms of a relatively small set of decimals in the range[1, 10).

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. Especially ifx is near 1, a good alternative is to useHalley's method orNewton's method to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value ofy to giveexp⁡(y)−x=0{\displaystyle \exp(y)-x=0} using Halley's method, or equivalently to giveexp⁡(y/2)−xexp⁡(−y/2)=0{\displaystyle \exp(y/2)-x\exp(-y/2)=0} using Newton's method, the iteration simplifies toyn+1=yn+2⋅x−exp⁡(yn)x+exp⁡(yn){\displaystyle y_{n+1}=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}}which hascubic convergence toln⁡(x){\displaystyle \ln(x)}.

Another alternative for extremely high precision calculation is the formula[14][15]ln⁡x≈π2M(1,4/s)−mln⁡2,{\displaystyle \ln x\approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,}whereM denotes thearithmetic-geometric mean of 1 and4/s, ands=x2m>2p/2,{\displaystyle s=x2^{m}>2^{p/2},}withm chosen so thatp bits of precision is attained. (For most purposes, the value of 8 form 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 constantsln⁡2{\displaystyle \ln 2} andπ can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used:ln⁡x=πM(θ22(1/x),θ32(1/x)),x∈(1,∞){\displaystyle \ln x={\frac {\pi }{M\left(\theta _{2}^{2}(1/x),\theta _{3}^{2}(1/x)\right)}},\quad x\in (1,\infty )}

whereθ2(x)=∑n∈Zx(n+1/2)2,θ3(x)=∑n∈Zxn2{\displaystyle \theta _{2}(x)=\sum _{n\in \mathbb {Z} }x^{(n+1/2)^{2}},\quad \theta _{3}(x)=\sum _{n\in \mathbb {Z} }x^{n^{2}}}are theJacobi theta functions.[16]

Based on a proposal byWilliam Kahan and first implemented in theHewlett-PackardHP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators,operating systems (for exampleBerkeley UNIX 4.3BSD[17]),computer algebra systems and programming languages (for exampleC99[18]) provide a specialnatural logarithm plus 1 function, alternatively namedLNP1,[19][20] orlog1p[18] to give more accurate results for logarithms close to zero by passing argumentsx, also close to zero, to a functionlog1p(x), which returns the valueln(1+x), instead of passing a valuey close to 1 to a function returningln(y).[18][19][20] The functionlog1p 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.[19][20]

In addition to basee, theIEEE 754-2008 standard defines similar logarithmic functions near 1 forbinary anddecimal logarithms:log2(1 +x) andlog10(1 +x).

Similar inverse functions named "expm1",[18] "expm"[19][20] or "exp1m" exist as well, all with the meaning ofexpm1(x) = exp(x) − 1.[nb 2]

An identity in terms of theinverse hyperbolic tangent,log1p(x)=log⁡(1+x)=2 artanh(x2+x),{\displaystyle \mathrm {log1p} (x)=\log(1+x)=2~\mathrm {artanh} \left({\frac {x}{2+x}}\right)\,,}gives a high precision value for small values ofx on systems that do not implementlog1p(x).

Computational complexity

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Main article:Computational complexity of mathematical operations

Thecomputational complexity of computing the natural logarithm using thearithmetic-geometric mean (for both of the above methods) isO(M(n)ln⁡n){\displaystyle {\text{O}}{\bigl (}M(n)\ln n{\bigr )}}. Here,n is the number of digits of precision at which the natural logarithm is to be evaluated, andM(n) is the computational complexity of multiplying twon-digit numbers.

Continued fractions

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While no simplecontinued fractions are available, severalgeneralized continued fractions exist, including:ln⁡(1+x)=x11−x22+x33−x44+x55−⋯=x1−0x+12x2−1x+22x3−2x+32x4−3x+42x5−4x+⋱{\displaystyle {\begin{aligned}\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^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\end{aligned}}}ln⁡(1+xy)=xy+1x2+1x3y+2x2+2x5y+3x2+⋱=2x2y+x−(1x)23(2y+x)−(2x)25(2y+x)−(3x)27(2y+x)−⋱{\displaystyle {\begin{aligned}\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{aligned}}}

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, thenatural logarithm of 2 can be computed as:ln⁡2=3ln⁡(1+14)+ln⁡(1+3125)=69−1227−2245−3263−⋱+6253−32759−621265−921771−⋱.{\displaystyle {\begin{aligned}\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{aligned}}}

Furthermore, since 10 = 1.2510 × 1.0243, even the natural logarithm of 10 can be computed similarly as:ln⁡10=10ln⁡(1+14)+3ln⁡(1+3125)=209−1227−2245−3263−⋱+18253−32759−621265−921771−⋱.{\displaystyle {\begin{aligned}\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{aligned}}}The reciprocal of the natural logarithm can be also written in this way:1ln⁡(x)=2xx2−112+x2+14x12+1212+x2+14x…{\displaystyle {\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 }

For example:1ln⁡(2)=4312+5812+1212+58…{\displaystyle {\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 }

Complex logarithms

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Main article:Complex logarithm

The exponential function can be extended to a function which gives acomplex number asez for any arbitrary complex numberz; simply use the infinite series withx=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: nox hasex = 0; and it turns out thate2iπ = 1 =e0. Since the multiplicative property still works for the complex exponential function,ez =ez+2kiπ, for all complexz and integers k.

So the logarithm cannot be defined for the wholecomplex plane, and even then it ismulti-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of2iπ at will. The complex logarithm can only be single-valued on thecut plane. For example,lni =⁠iπ/2⁠ or⁠5iπ/2⁠ or−⁠3iπ/2⁠, etc.; and althoughi4 = 1, 4 lni can be defined as2iπ, or10iπ or−6iπ, and so on.

  • Plots of the natural logarithm function on the complex plane (principal branch)
  • z = Re(ln(x + yi))
    z = Re(ln(x +yi))
  • z = |(Im(ln(x + yi)))|
    z = |(Im(ln(x +yi)))|
  • z = |(ln(x + yi))|
    z = |(ln(x +yi))|
  • Superposition of the previous three graphs
    Superposition of the previous three graphs

See also

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Notes

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  1. ^ IncludingC,C++,Java,SAS,MATLAB,Mathematica,Fortran, and someBASIC dialects
  2. ^For a similar approach to reduceround-off errors of calculations for certain input values seetrigonometric functions likeversine,vercosine,coversine,covercosine,haversine,havercosine,hacoversine,hacovercosine,exsecant andexcosecant.

References

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  1. ^Sloane, N. J. A. (ed.)."Sequence A001113 (Decimal expansion of e)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^Hardy, G.H.; Wright, E.M. (1975).An Introduction to the Theory of Numbers (4th ed.). Oxford. footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest".
  3. ^Mortimer, Robert G. (2005).Mathematics for physical chemistry (3rd ed.).Academic Press. p. 9.ISBN 0-12-508347-5.Extract of page 9
  4. ^abcWeisstein, Eric W."Natural Logarithm".mathworld.wolfram.com. Retrieved2020-08-29.
  5. ^ab"Rules, Examples, & Formulas". Logarithm.Encyclopedia Britannica. Retrieved2020-08-29.
  6. ^Burn, R.P. (2001). "Alphonse Antonio de Sarasa and logarithms".Historia Mathematica.28:1–17.doi:10.1006/hmat.2000.2295.
  7. ^O'Connor, J. J.; Robertson, E. F. (September 2001)."The number e". The MacTutor History of Mathematics archive. Retrieved2009-02-02.
  8. ^abCajori, Florian (1991).A History of Mathematics (5th ed.). AMS Bookstore. p. 152.ISBN 0-8218-2102-4.
  9. ^""Logarithmic Expansions" at Math2.org".
  10. ^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
  11. ^RUFFA, Anthony."A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES"(PDF).International Journal of Mathematics and Mathematical Sciences. Retrieved2022-02-27. (Page 3654, equation 2.6)
  12. ^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.
  13. ^Sloane, N. J. A. (ed.)."Sequence A002392 (Decimal expansion of natural logarithm of 10)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. ^Sasaki, T.; Kanada, Y. (1982)."Practically fast multiple-precision evaluation of log(x)".Journal of Information Processing.5 (4):247–250. Retrieved2011-03-30.
  15. ^Ahrendt, Timm (1999). "Fast Computations of the Exponential Function".Stacs 99. Lecture Notes in Computer Science. Vol. 1564. pp. 302–312.doi:10.1007/3-540-49116-3_28.ISBN 978-3-540-65691-3.
  16. ^Borwein, Jonathan M.; Borwein, Peter B. (1987).Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.ISBN 0-471-83138-7. page 225
  17. ^Beebe, Nelson H. F. (2017-08-22). "Chapter 10.4. Logarithm near one".The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA:Springer International Publishing AG. pp. 290–292.doi:10.1007/978-3-319-64110-2.ISBN 978-3-319-64109-6.LCCN 2017947446.S2CID 30244721.In 1987, Berkeley UNIX 4.3BSD introduced the log1p() function
  18. ^abcdBeebe, Nelson H. F. (2002-07-09)."Computation of expm1 = exp(x)−1"(PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing,University of Utah. Retrieved2015-11-02.
  19. ^abcdHP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.).Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved2015-09-06.
  20. ^abcdHP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.).Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved2015-10-10.Searchable PDF
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