Answer
$$
y=f(x)=3^{2 x-4}
$$
The inverse function $f^{-1}(x)$ is given by:
$$
f^{-1}(x)=\frac{1}{2} \log _{3} x+2
$$
Work Step by Step
$$
y=f(x)=3^{2 x-4}
$$
We take logarithms of both sides of the equation and use (6):
$$
\log _{3} y=\log _{3} \left[ 3^{2 x-4}\right] =(2 x-4)\log _{3} \left[ 3\right] =(2 x-4)
$$
$\Rightarrow$
$$
2 x=\log _{3} y+4
$$
$\Rightarrow $
$$
x=\frac{1}{2} \log _{3} y+2 .
$$
Interchange $x$ and $y$ :
$$
y=\frac{1}{2} \log _{3} x+2 .
$$
So the inverse function $f^{-1}(x)$ is given by:
$$
f^{-1}(x)=\frac{1}{2} \log _{3} x+2
$$
