Chapter 11.5, Problem 31E

To determine

To calculate:

Inverse of the given function.

Answer to Problem 31E

Inverse of the function $y={\log }_{6}x-1$ is $f^{-1}\left(x\right)=6^{x+1}$ .

Explanation of Solution

Given information:

  $y={\log }_{6}x-1$

Calculation:

It is given that,

  $y={\log }_{6}x-1$

Now interchange $x$ and $y$ ,

  $x={\log }_{6}y-1$

Add $1$ to both sides and simplify,

  $x+1={\log }_{6}y$

Rewrite as exponential form,

  $6^{x-1}=6^{{\log }_{6}y}$

Now simplify,

  $6^{x-1}=y$

  $f^{-1}\left(x\right)=6^{x+1}$

Hence, Inverse of the given function is $f^{-1}\left(x\right)=6^{x+1}$ .