Answer
$g^{-1}(x)= \frac{\log \left(- \frac{x}{3} \right)}{\log 5}$
Work Step by Step
Given \begin{equation}
g(x)=-3(5)^x.
\end{equation} Let $y= -3(5)^x $. Solve for $x$ in terms of $y$.
\begin{equation}
\begin{aligned}
-3(5)^x &= y\\
3(5)^x &= -y\\
\log\left(3(5)^x\right) & = \log (-y)\\
\log (5)^x+\log(3) & = \log (-y)\\
x\log 5&= \log (-y)-\log(3)\\
x\log 5&= \log \left(- \frac{y}{3} \right)\\
\therefore x&= \frac{\log \left(- \frac{y}{3} \right)}{\log 5}.
\end{aligned}
\end{equation} Interchange $x$ and $y$. That is, set $y= \frac{\log \left(- \frac{x}{3} \right)}{\log 5}$. The inverse of the function is: \begin{equation}
g^{-1}(x)= \frac{\log \left(- \frac{x}{3} \right)}{\log 5}.
\end{equation}
