#### Answer

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

#### Work Step by Step

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