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Natural Logarithm

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NaturalLogarithm

The natural logarithm lnx is the logarithm having base e, where

e=2.718281828....
(1)

This function can be defined

lnx=int_1^x(dt)/t
(2)

for x>0.

NaturalLogEPlot

This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola y=1/x, the x-axis, and the vertical lines x=1 and x=e is 1. In other words,

int_1^e(dx)/x=lne=1.
(3)

The notation lnx is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation logx. In this work, lnx=log_ex denotes a natural logarithm, whereas logx=log_(10)x denotes the common logarithm.

There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use ln^nz (i.e., using a trigonometric function-like convention), it is also common to write (lnz)^n.

Common and natural logarithms can be expressed in terms of each other as

lnx=(log_(10)x)/(log_(10)e)
(4)
log_(10)x=(lnx)/(ln10).
(5)

The natural logarithm is especially useful in calculus because its derivative is given by the simple equation

d/(dx)lnx=1/x,
(6)

whereas logarithms in other bases have the more complicated derivative

d/(dx)log_bx=1/(xlnb).
(7)
NaturalLogBranchCut

The natural logarithm can be analytically continued to complex numbers as

lnz=ln|z|+iarg(z),
(8)

where |z| is the complex modulus and arg(z) is the complex argument. The natural logarithm is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at (-infty,0].

NaturalLogReImAbs
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The principal value of the natural logarithm is implemented in the Wolfram Language as Log[x], which is equivalent to Log[E, x]. This function is illustrated above in the complex plane.

Note that the inverse trigonometric and inverse hyperbolic functions can be expressed (and, in fact, are commonly defined) in terms of the natural logarithm, as summarized in the table below. Therefore, once these definition are agreed upon, the branch cut structure adopted for the natural logarithm fixes the branch cuts of these functions.

functionsymboldefinition
inverse cosecantcsc^(-1)z-iln(sqrt(1-1/(z^2)+i/z))
inverse cosinecos^(-1)z1/2pi+iln(iz+sqrt(1-z^2))
inverse cotangentcot^(-1)z1/2i[ln((z-i)/z)-ln((z+i)/z)]
inverse hyperbolic cosecantcsch^(-1)zln(sqrt(1+1/(z^2))+1/z)
inverse hyperbolic cosinecosh^(-1)zln(z+sqrt(z-1)sqrt(z+1))
inverse hyperbolic cotangentcoth^(-1)z1/2[ln(1+1/z)-ln(1-1/z)]
inverse hyperbolic secantsech^(-1)zln(sqrt(1/z-1)sqrt(1/z+1)+1/z)
inverse hyperbolic sinesinh^(-1)zln(z+sqrt(z^2+1))
inverse hyperbolic tangenttanh^(-1)z1/2[ln(1+z)-ln(1-z)]
inverse secantsec^(-1)z1/2pi+iln(sqrt(1-1/(z^2))+i/z)
inverse sinesin^(-1)z-iln(iz+sqrt(1-z^2))
inverse tangenttan^(-1)z1/2i[ln(1-iz)-ln(1+iz)]

The Mercator series

ln(1+x)=x-1/2x^2+1/3x^3-...
(9)

gives a Taylor series for the natural logarithm.

Continued fraction representations of logarithmic functions include

ln(1+x)=x/(1+(1^2x)/(2+(1^2x)/(3+(2^2x)/(4+(2^2x)/(5+(3^2x)/(6+(3^2x)/(7+...)))))))
(10)

(Lambert 1770; Lagrange 1776; Olds 1963, p. 138; Wall 1948, p. 342) and

ln((1+x)/(1-x))=(2x)/(1-(x^2)/(3-(4x^2)/(5-(9x^2)/(7-(16x^2)/(9-...)))))
(11)

(Euler 1813-1814; Wall 1948, p. 343; Olds 1963, p. 139).

For a complex number z, the natural logarithm satisfies

lnz=ln[re^(i(theta+2npi))]
(12)
=lnr+i(theta+2npi)
(13)

and

PV(lnz)=lnr+itheta,
(14)

where PV is the principal value.

Some special values of the natural logarithm include

ln(-1)=ipi
(15)
ln0=-infty
(16)
ln1=0
(17)
lne=1
(18)
ln(+/-i)=+/-1/2pii.
(19)

Natural logarithms can sometimes be written as a sum or difference of "simpler" logarithms, for example

ln(2+sqrt(3))=2ln(1+sqrt(3))-ln2,
(20)

which follows immediately from the identity

2+sqrt(3)=1/2(1+sqrt(3))^2.
(21)

Plouffe (2006) found the following beautiful identities:

ln2=10sum_(n=1)^(infty)1/(n(e^(pin)+1))+6sum_(n=1)^(infty)1/(n(e^(pin)-1))-4sum_(n=1)^(infty)1/(n(e^(2pin)-1))
(22)
ln3=9sum_(n=1)^(infty)1/(n(e^(pin)-1))+(49)/3sum_(n=1)^(infty)1/(n(e^(pin)+1))-(14)/3sum_(n=1)^(infty)1/(n(e^(2pin)+1))+sum_(n=1)^(infty)1/(n(e^(3pin)-1))-7/3sum_(n=1)^(infty)1/(n(e^(3pin)+1))+2/3sum_(n=1)^(infty)1/(n(e^(6pin)+1))
(23)
ln5=(57)/4sum_(n=1)^(infty)1/(n(e^(pin)-1))+(91)/4sum_(n=1)^(infty)1/(n(e^(pin)+1))-(13)/2sum_(n=1)^(infty)1/(n(e^(2pin)+1))+3/4sum_(n=1)^(infty)1/(n(e^(5pin)-1))-7/4sum_(n=1)^(infty)1/(n(e^(5pin)+1))+1/2sum_(n=1)^(infty)1/(n(e^(10pin)+1)).
(24)

See also

Common Logarithm, e, Lg, Logarithm, Nat, Natural Logarithm Catacaustic, Natural Logarithm of 2, Natural Logarithm of 10 Explore this topic in the MathWorld classroom

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References

Euler, L. "Commentatio in fractionem continuam qua illustris La Grange potestates binomiales expressit." Mém. de l'Acad. imperiale des sciences de St. Pétersbourg 6, 1813-1814.Lagrange, J.-L. "Sur l'usage des fractions continues dans le calcul intégral." Nouv. mém. de l'académie royale des sciences et belles-lettres Berlin, 236-264, 1776. Reprinted in Oeuvres, Vol. 4, pp. 301-302.Lambert, J. L. Beiträge zum Gebrauch der Mathematik und deren Anwendung. Theil 2. Berlin, 1770.Olds, C. D. Continued Fractions. New York: Random House, 1963.Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, 1996.

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Natural Logarithm

Cite this as:

Weisstein, Eric W. "Natural Logarithm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithm.html

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