Find the Inverse 1/3 log of 4x

Rewrite the equation as ${\displaystyle \frac{1}{3}\cdot \log \left(4y\right)=x}$.

Multiply both sides of the equation by ${\displaystyle 3}$.

Simplify the left side.

Rewrite ${\displaystyle \log \left(4y\right)=3x}$ in exponential form using the definition of a logarithm. If ${\displaystyle x}$ and ${\displaystyle b}$ are positive real numbers and ${\displaystyle b\ne 1}$, then ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$.

Solve for ${\displaystyle y}$.

To verify the inverse, check if ${\displaystyle f^{-1}\left(f\left(x\right)\right)=x}$ and ${\displaystyle f\left(f^{-1}\left(x\right)\right)=x}$.

Evaluate ${\displaystyle f^{-1}\left(f\left(x\right)\right)}$.

Evaluate ${\displaystyle f\left(f^{-1}\left(x\right)\right)}$.

Since ${\displaystyle f^{-1}\left(f\left(x\right)\right)=x}$ and ${\displaystyle f\left(f^{-1}\left(x\right)\right)=x}$, then ${\displaystyle f^{-1}\left(x\right)=\frac{{10}^{3x}}{4}}$ is the inverse of ${\displaystyle f\left(x\right)=\frac{1}{3}\cdot \log \left(4x\right)}$.