Question

How to find the inverse of 3log3(x+3)+1. Its with a base of 3

Answer 1

Let
`f^(-1)(x)` be the inverse of

`f(x) = 3 \log_3 (x + 3) + 1`

Then by definition of inverse function,

`f\left( f^(-1)(x) \right) = 3 \log_3 \left(f^(-1)(x) + 3\right) + 1 = x`

Solve for the inverse :

`3 \log_3 \left(f^(-1)(x) + 3\right) + 1 = x`

`3 \log_3 \left(f^(-1)(x) + 3\right) = x - 1`

`\log_3 \left(f^(-1)(x) + 3\right) = \frac{x - 1}3`

`3^{\log_3 \left(f^(-1)(x) + 3\right)} = 3^{\frac{x-1}3}`

`f^(-1)(x) + 3 = 3^{\frac{x-1}3}`

`f^(-1)(x) = 3^{\frac{x-1}3} - 3`