Question

What is the inverse of `y = log_3(x-2)` ?

Answer 1

Inverse to `f(x)=log_3(x-2)` is `g(x)=3^x+2`.

Explanation:

Function `y=f(x)` is inverse to `y=g(x)` if and only if the composition of these function is an identity function `y=x`.

The function we have to inverse is `f(x)=log_3(x-2)`
Consider function `g(x)=3^x+2`.

The composition of these functions is:
`f(g(x)) = log_3(3^x+2-2) = log_3(3^x) = x`

The other composition of the same functions is
`g(f(x)) = 3^(log_3(x-2))+2 = x-2+2 = x`

As you see, inverse to `f(x)=log_3(x-2)` is `g(x)=3^x+2`.