Suppose \(f(x)=3\cdot 5^{x+2}\text{.}\) We want to find a formula for \(f^{-1}(x)\text{.}\) Recall that to find the inverse of a function, we start by writing \(y=f(x)\text{,}\) then swap \(x\) and \(y\text{,}\) then solve for \(y\text{.}\)
Our equation starts as
\begin{equation*}
y=3\cdot 5^{x+2}
\end{equation*}
\begin{equation*}
x=3\cdot 5^{y+2}
\end{equation*}
Now we need to solve this equation for \(y\) using what we know about solving exponential equations.
\begin{align*}
x\amp=3\cdot 5^{y+2}\\
\frac{x}{\color{red}{3}}\amp=\frac{3\cdot 5^{y+2}}{\color{red}{3}}\\
\frac{x}{3}\amp= 5^{y+2}\\
{\color{red}{\log_5}}\left(\frac{x}{3}\right)\amp= {\color{red}{\log_5}}\left(5^{y+2}\right)\\
\log_5\left(\frac{x}{3}\right)\amp= y+2\\
\log_5\left(\frac{x}{3}\right){\color{red}{-2}}\amp= y+2{\color{red}{-2}}\\
\log_5\left(\frac{x}{3}\right)-2\amp= y
\end{align*}
Therefore, our answer is that \(f^{-1}(x)=\log_5\left(\frac{x}{3}\right)-2\text{.}\)