Find the inverse f(x)={e}^(3x - 1) Tank you for your support ​

Question

Find the inverse

`f(x)={e}^(3x - 1)`
Tank you for your support ​

Answer 1

`f^(-1)(x)=(1)/(3)(\ln x +1)`

Explanation:

Given function:

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

Replace f(x) with y:

`\implies y=e^(3x-1)`

Take natural logs of both sides:

`\implies \ln y = \ln e^(3x-1)`

Apply the Power Log Law
`\ln x^n=n\ln x` :

`\implies \ln y = (3x-1)\ln e`

As
`\ln e=1` then:

`\implies \ln y = 3x-1`

Rearrange to make x the subject:

`\implies 3x=\ln y +1`

`\implies x=(1)/(3)(\ln y +1)`

Swap x for
`f^(-1)(x)` and y for x:

`\implies f^(-1)(x)=(1)/(3)(\ln x +1)`