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).