# How do you find the inverse of y=e^x?

Jan 15, 2016

The inverse function is $\ln x$.

#### Explanation:

By definition, $y = {f}^{- 1} \left(x\right) \iff f \left(y\right) = x$

$\iff {e}^{y} = x$

$\iff y = \ln x$.

Jan 15, 2016

$x = \ln \left(y\right)$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} y = {e}^{x}$

Taking the natural logarithm of both sides
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\ln \left(y\right) = \ln \left({e}^{x}\right)}$

By definition of $\ln$
$\textcolor{w h i t e}{\text{XXX}} \ln \left(a\right) =$ the value. $b$, needed to make ${e}^{b} = a$
(you should memorize this)
therefore
$\textcolor{w h i t e}{\text{XXX}} \ln \left({e}^{x}\right) =$ the value,$b$, needed to make ${e}^{b} = {e}^{x}$
that is
$\textcolor{w h i t e}{\text{XXX}} \ln \left({e}^{x}\right) = x$

Therefore $\textcolor{red}{\ln \left(y\right) = \ln \left({e}^{x}\right)}$
implies
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{x = \ln \left(y\right)}$