# What is the inverse of y = log_2(x^2) ?

Dec 18, 2015

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

#### Explanation:

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

The logarithm of the second power of a number is twice the logarithm of the number itself:

$\implies y = \textcolor{red}{2} {\log}_{2} x$
$\implies \textcolor{red}{\frac{1}{2} \times} y = \textcolor{red}{\frac{1}{2} \times} 2 {\log}_{2} x$
$\implies x = {2}^{\frac{y}{2}}$

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