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

Feb 17, 2016

Inverse to $f \left(x\right) = {\log}_{3} \left(x - 2\right)$ is $g \left(x\right) = {3}^{x} + 2$.

Explanation:

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

The function we have to inverse is $f \left(x\right) = {\log}_{3} \left(x - 2\right)$
Consider function $g \left(x\right) = {3}^{x} + 2$.

The composition of these functions is:
$f \left(g \left(x\right)\right) = {\log}_{3} \left({3}^{x} + 2 - 2\right) = {\log}_{3} \left({3}^{x}\right) = x$

The other composition of the same functions is
$g \left(f \left(x\right)\right) = {3}^{{\log}_{3} \left(x - 2\right)} + 2 = x - 2 + 2 = x$

As you see, inverse to $f \left(x\right) = {\log}_{3} \left(x - 2\right)$ is $g \left(x\right) = {3}^{x} + 2$.