Find the Inverse y=-3 log base 6 of x

Rewrite the equation as ${\displaystyle -3{\log }_6\left(y\right)=x}$.

Divide each term in ${\displaystyle -3{\log }_6\left(y\right)=x}$ by ${\displaystyle -3}$ and simplify.

Rewrite ${\displaystyle {\log }_6\left(y\right)=-\frac{x}{3}}$ 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 ${\displaystyle b^y=x}$.

Rewrite the equation as ${\displaystyle y=6^{-\frac{x}{3}}}$.

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 ${\displaystyle f^{-1}\left(x\right)=6^{-\frac{x}{3}}}$ is the inverse of ${\displaystyle f\left(x\right)=-3{\log }_6\left(x\right)}$.