Answer
$g^{-1}(x) = \log_3 \frac{x}{4}$
Work Step by Step
$g(x) = 4(3)^{x}$
Let $g(x) = y$
$y = 4(3)^{x}$
Swap $x$ and $y$ then solve for $y$
$x = 4(3)^{y}$
$\frac{x}{4} = (3)^{y}$
$y = \log_3 \frac{x}{4}$
$g^{-1}(x) = \log_3 \frac{x}{4}$