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 :

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

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

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

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

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

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