Question

How do you find the inverse of `y=log_3 9x`?

Answer 1

`f^-1(x)=3^(x-2)`

Explanation:

Given `y=log_3(9x)`

We know that if `f(x)=y` then `f^-1(y)=x`

So, from the above equation, by separating the log into 2 values, we get `log_3(9x)=log_3(9)+log_3x`
Since `9=3^2`, the first term in the log value becomes `log_3(9)=log_3(3^2)=2log_3(3)=2`

So, `y=2+log_3(x)\impliesy-2=log_3x`

I believe you're familiar with this general log equation,
`log_nm=p\impliesm=n^p`

So, the above equation becomes `3^(y-2)=x`
Taking for a fact that f^-1(y)=x#, we see what the inverse of the original function is.