Find the Inverse y = log base 8 of x

Rewrite the equation as ${\displaystyle {\log }_8\left(y\right)=x}$.

Rewrite ${\displaystyle {\log }_8\left(y\right)=x}$ in exponential form using the definition of a logarithm. If ${\displaystyle x}$ and ${\displaystyle b}$ are positive real numbers and ${\displaystyle b}$${\displaystyle \ne }$${\displaystyle 1}$, then ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$.

Rewrite the equation as ${\displaystyle y=8^x}$.

Replace the ${\displaystyle y}$ with ${\displaystyle f^{-1}\left(x\right)}$ to show the final answer.

Set up the composite result function.

Evaluate ${\displaystyle g\left(f\left(x\right)\right)}$ by substituting in the value of ${\displaystyle f}$ into ${\displaystyle g}$.

Exponentiation and log are inverse functions.

Since ${\displaystyle g\left(f\left(x\right)\right)=x}$, ${\displaystyle f^{-1}\left(x\right)=8^x}$ is the inverse of ${\displaystyle f\left(x\right)={\log }_8\left(x\right)}$.