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# Natural logarithm function

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
${\displaystyle \log e^{x}:=\log _{e}e^{x}:=x.\,}$
Base
 e
is considered the “natural” base since we have
${\displaystyle {\frac {d\,a^{x}}{dx}}={\frac {d\,(e^{\log a})^{x}}{dx}}={\frac {d\,{e^{x}}^{\log a}}{dx}}=(\log a){\frac {d\,e^{x}}{dx}}=(\log a)\,e^{x},\,}$
where
 e  x
is the eigenfunction of the derivative, i.e.
${\displaystyle {\frac {d\,e^{x}}{dx}}=e^{x}.\,}$
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).