# What is the inverse of f(x) = 2^x ?

Dec 23, 2015

$\textcolor{w h i t e}{\times} {f}^{-} 1 \left(x\right) = {\log}_{2} x$

#### Explanation:

$\textcolor{w h i t e}{\times} f \left(x\right) = {2}^{x}$

$\implies y = {\textcolor{red}{2}}^{x} \textcolor{w h i t e}{\times \times \times \times \times x}$ (base is $\textcolor{red}{2}$)
$\implies x = {\log}_{\textcolor{red}{2}} y \textcolor{w h i t e}{\times \times \times \times \times x}$(logarithm definition)

$\implies {f}^{-} 1 \left(x\right) = {\log}_{2} x$

In ${\mathbb{R}}^{2}$, ${f}^{-} 1 \left(x\right)$ graph have to be symmetrical of $f \left(x\right)$ graph:

$y = f \left(x\right)$, $y = x$, and $y = {f}^{-} 1 \left(x\right)$ graphs