# What is the inverse of a logarithmic function?

Dec 10, 2015

The inverse of a logarithmic function is exponential function.

#### Explanation:

The inverse of a logarithmic function is exponential function:
$\textcolor{w h i t e}{\times x} {f}^{-} 1 \left(x\right) = {g}^{-} 1 \left({a}^{x}\right)$
Because logarithmic fuction is
$\textcolor{w h i t e}{\times x} f \left(x\right) = {\log}_{a} g \left(x\right)$
$\implies g \left(x\right) = {a}^{f} \left(x\right)$
$\implies {g}^{-} 1 \left(g \left(x\right)\right) = {g}^{-} 1 \left({a}^{f} \left(x\right)\right)$
$\implies x = {g}^{-} 1 \left({a}^{f} \left(x\right)\right)$
$\implies {f}^{-} 1 \left(x\right) = {g}^{-} 1 \left({a}^{x}\right)$

Considering f(x)=x is axis of symmetry, for $g \left(x\right) = x$ and $a = 10$,