Chapter 4, Exponential and Logarithmic Functions - Section 4.5 - Exponential and Logarithmic Functions - 4.5 Exercises - Page 405: 84

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

We are given: $f(x)=3^{x+1}$ To find the inverse, we switch $x$ and $y$ and solve for $y=3^{x+1}$ $x=3^{y+1}$ $\ln x=\ln(3^{y+1})$ $\ln x=(y+1)\ln 3$ $y+1= \frac{\ln x}{\ln 3}$ $y=\frac{\ln x}{\ln 3}-1$ Therefore: $f^{-1}(x)= \frac{\ln x}{\ln 3}-1$