Chapter 6.3, Problem 47E

To determine

To calculate: The inverse of the function.

Answer to Problem 47E

The inverse of the function $y=\ln \left(x-1\right)$ is $e^y=x-1$ .

Explanation of Solution

Given information:

The given function is $y=\ln \left(x-1\right)$ .

Formula Used:

Logarithmic power rule is

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

Calculation:

Consider the logarithmic function is $y=\ln \left(x-1\right)$ .

Convert the logarithmic form into exponential form.

Since the given function is a natural logarithmic form thus, the base of the function is $e$ .

Take exponential in each side with base $e$ . Use the power rule.

  $\begin{array}{c}y=\ln \left(x-1\right)\\ e^y=e^{\ln \left(x-1\right)}\\ e^y=x-1\end{array}$

Hence, the inverse of the function $y=\ln \left(x-1\right)$ is $e^y=x-1$ .