Inverse function

I attempted to find the inverse function of y=2lnx-2ln(x-1) but I think my answer is wrong I have uploaded a pic below. Also the domain of the inverse function is y>0 because it’s a exponential function so y can’t be 0?

I guess a couple things,

The domain of an exponential is R. For the inverse function, you could tink about what happens when y=0. But you may want to sketch the two functions and think about the start/end values in each case. You could note that the inverse function could be written as
1 + 1/(e^(y/2)-1)
which is similar to a reciprocal.