Question

If `f(x) = 3^x`, what is `f^-1(x)`?

Answer 1

The inverse function is `f^-1(x)=lnx/ln3`

Explanation:

Let `y=3^x`

To calculate the inverse, interchange `x` and `y`,

`x=3^y`

Express `y` in terms of `x`

`lnx=ln(3^y)`

`lnx=yln3`

`y=lnx/ln3`

Therefore,

`f^-1(x)=lnx/ln3`

Verification :

`f(f^-1(x))=f(lnx/ln3)=3^(lnx/ln3)`

Let `y=3^(lnx/ln3)`

`=>`

`lny=lnx/ln3*ln3=lnx`

`y=x`

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

So, the functions `3^x` and `lnx/ln3` are inverses