6.3 Logarithmic functions  (Page 4/9)

Definition of the natural logarithm

A natural logarithm    is a logarithm with base $e.$ We write ${\log }_e\left(x\right)$ simply as $\ln \left(x\right).$ The natural logarithm of a positive number $x$ satisfies the following definition.

For $x>0,$

We read $\ln \left(x\right)$ as, “the logarithm with base $e$ of $x$ ” or “the natural logarithm of $x.$ ”

The logarithm $y$ is the exponent to which $e$ must be raised to get $x.$

Since the functions $y=e{}^x$ and $y=\ln \left(x\right)$ are inverse functions, $\ln \left(e^x\right)=x$ for all $x$ and $e{}^{\ln (x)}=x$ for $x>0.$