Natural logarithm function

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The natural logarithm function is the logarithm with base

e

, i.e. Euler’s number. It is the inverse function of the [natural] exponential function, also with base

e

, thus

$\log e^x:={\log }_e{e}^x:=x.$

Base

e

is considered the “natural” base since we have

$\frac{d{a}^x}{dx}=\frac{d(e^{\log a}{)}^x}{dx}=\frac{d{e^x}^{\log a}}{dx}=(\log a)\frac{d{e}^x}{dx}=(\log a)e^x,$

where

e  x

is the eigenfunction of the derivative, i.e.

$\frac{d{e}^x}{dx}=e^x.$

In some scientific or engineering contexts, the notation

log x

may signify the common logarithm

log10 x

, in which case the natural logarithm is denoted

ln x

.

Applications

The natural log/e has ties to numerous applications, only a few of which include statistics (Poisson distribution), number theory (Prime number theorem), and thermodynamic (entropy).