The given function is
[tex]f(x)=\ln 5x[/tex]It is important to know that the inverse function for logarithms is exponential functions. In this case, we change f(x) to y.
[tex]y=\ln 5x[/tex]Then, we isolate x
[tex]\begin{gathered} e^y=e^{\ln 5x} \\ e^y=5x \\ x=\frac{e^y}{5} \end{gathered}[/tex]At last, we change the variables.
[tex]y=\frac{e^x}{5}[/tex]The inverse function notation is
[tex]f^{-1}(x)=\frac{e^x}{5}^{}[/tex]