Chapter 6.3, Problem 52E

To determine

To calculate: The inverse of the function.

Answer to Problem 52E

The inverse of the function $y=13+\log x$ is $x={10}^{\left(y-13\right)}$ .

Explanation of Solution

Given information:

The given function is $y=13+\log x$ .

Formula Used:

Logarithmic power rule is

If ${\log }_{a}b$ is a logarithmic function then the exponential form $a^{{\log }_{a}b}$ is equal to $b$ .

Calculation:

Consider the logarithmic function is $y=13+\log x$ .

Rewrite the equation as $y=13+\log x$ .

Convert the logarithmic form into exponential form.

Since the given logarithmic function is in common log form. Thus, the base of the log is 10.

Take exponential in each side with base 10.

  $\begin{array}{c}y-13=\log x\\ {10}^{\left(y-13\right)}={10}^{\log x}\\ {10}^{\left(y-13\right)}=x\\ x={10}^{\left(y-13\right)}\end{array}$

Hence, the inverse of the function $y=13+\log x$ is $x={10}^{\left(y-13\right)}$ .