What is the inverse of f(x)=ln(x-6)

Solve
y = ln(x - 6)
e^y = x - 6  Take the inverse. Interchange y and x
e^x = y - 6
y = e^x + 6

Remark
One of the ways you can check you answer is to see if the two graphs (the original and the inverse) are symmetrical about the line of y = x. What that means is that if you stuck a mirror on y = x, the reflection would look like a mirror image.

In this case The red line looks like the mirror image of the blue line and the green line is y = x where the mirror goes.