Chapter 4, Exponential and Logarithmic Functions - Section 4.5 - Exponential and Logarithmic Functions - 4.5 Exercises - Page 405: 83

$f^{-1}(x)= \frac{\ln x}{2\ln 2}$

We are given: $f(x)=2^{2x}$ To find the inverse, we switch $x$ and $y$ and solve for $y$: $y=2^{2x}$ $x=2^{2y}$ $\ln x=\ln 2^{2y}$ $\ln x=2y \ln 2$ $y=\frac{\ln x}{2\ln 2}$ Therefore: $f^{-1}(x)= \frac{\ln x}{2\ln 2}$