Answer
$g^{-1} (x) = \ln \frac{x}{2}$
Work Step by Step
$g(x) = 2e^{x}$
Let $g(x) = y$
$y = 2e^{x}$
Swap the variables $x$ and $y$ and then solve for $y$ to find the inverse:
$x = 2e^{y}$
$\frac{x}{2} = e^{y}$
$\ln \frac{x}{2} = y$
$g^{-1} (x) = \ln \frac{x}{2}$
