The natural logarithm is the logarithm having base e, where

This function can be defined

for .

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

The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas 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 (i.e., using a trigonometric function-like convention), it is also common to write .

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

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

whereas logarithms in other bases have the more complicated derivative

The natural logarithm can be analytically continued to complex numbers as

where is the complex modulus and 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 .

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.

The Mercator series

gives a Taylor series for the natural logarithm.

Continued fraction representations of logarithmic functions include

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

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

For a complex number , the natural logarithm satisfies

and

where is the principal value.

Some special values of the natural logarithm include

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

which follows immediately from the identity

Plouffe (2006) found the following beautiful identities:

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 Resource. https://mathworld.wolfram.com/NaturalLogarithm.html

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