Chapter 6.3, Problem 44E

To determine

To calculate: The inverse of the function.

Answer to Problem 44E

The inverse of the function $y={11}^x$ is ${\log }_{11}y=x$ .

Explanation of Solution

Given information:

The given function is $y={11}^x$ .

Formula Used:

Logarithmic power rule is

If $e^b$ is an expression then logarithm form ${\log }_a{\left(a\right)}^b$ is equal to $b$ .

Calculation:

Consider the exponential function is $y={11}^x$ .

Convert the exponential form into logarithmic form.

Take logarithm in each side. Use the logarithmic power rule.

  $\begin{array}{l}\log y=\log {11}^x\\ \log y=x\times \log 11\end{array}$

It is known that for two log function $\log a\text{and}\log b$ , their division $\frac{\log a}{\log b}={\log }_{b}a$ .

Now, solve the above expression, use the formula $\frac{\log a}{\log b}={\log }_{b}a$ .

  $\begin{array}{l}\log y=x\times \log 11\\ x=\frac{\log y}{\log 11}\\ x={\log }_{11}y\end{array}$

Hence, the inverse of the function $y={11}^x$ is ${\log }_{11}y=x$ .