# Inverse of e to the power 2x?

Apr 20, 2018

$y = \ln \frac{x}{2}$

#### Explanation:

So, we have

$y = {e}^{2 x}$ and want to find its inverse. As a result, we'll switch the places of $x , y$ and subsequently solve for $y :$

$x = {e}^{2 y}$

Applying the logarithm to both sides:

$\ln \left({e}^{2 y}\right) = \ln x$

Recalling that $\ln \left({a}^{b}\right) = b \ln a ,$

$2 y \ln e = \ln x$

$\ln \left(e\right) = 1 ,$ so

$2 y = \ln x$

$y = \ln \frac{x}{2}$