Answer
$f^{-1}(x)=e^{\frac{x}{2}}+1$
Work Step by Step
Given: $y=2ln(x-1)$
$ln(x-1)=\frac{y}{2}$
$x-1=e^{\frac{y}{2}}$
$x=e^{\frac{y}{2}}+1$
Now, replace the terms to find out the value of inverse function.
$y=e^{\frac{x}{2}}+1$
Hence, $f^{-1}(x)=e^{\frac{x}{2}}+1$