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The

**natural logarithm function** is the logarithm with base

, i.e.

Euler’s number. It is the inverse function of the [natural]

exponential function, also with base

, thus

- $\log e^{x}:=\log _{e}e^{x}:=x.\,$

Base

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

is the eigenfunction of the derivative, i.e.

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

In some scientific or engineering contexts, the notation

may signify the

common logarithm , in which case the natural logarithm is denoted

.

## 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).