Chapter 4, Exponential and Logarithmic Functions - Section 4.5 - Exponential and Logarithmic Functions - 4.5 Exercises - Page 405: 85

$f^{-1}(x)=2^{x}+1$

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