Chapter 6.6, Problem 22E

To determine

To calculate: .The equation for the value of $x$ .

Answer to Problem 22E

The required value of $x$ is $10$ .

Explanation of Solution

Given Information:

The given equation is $\ln \left(2x-4\right)=\ln \left(x+6\right)$ .

Formula Used:

Use property of logarithms:

  ${\log }_a\left(x\right)=a^{{\log }_a\left(x\right)}$

Cancellation rule of logarithms:

  $a^{{\log }_a\left(x\right)}=x$

Calculation:

Consider the equation is $\ln \left(2x-4\right)=\ln \left(x+6\right)$ .

It is known that the base of a natural log is $e$ .

Rewrite the given equation with base $e$ .

  ${\ln }_e\left(2x-4\right)={\ln }_e\left(x+6\right)$

Use the property of logarithms ${\log }_a\left(x\right)=a^{{\log }_a\left(x\right)}$ of the above equation.

  $e^{{\ln }_e\left(2x-4\right)}=e^{{\ln }_e\left(x+6\right)}$

Use the cancellation rule of logarithms $a^{{\log }_a\left(x\right)}=x$ of the above equations.

  $2x-4=x+6$

Simplify the above equation.

  $\begin{array}{c}2x-4=x+6\\ 2x-x=6+4\\ x=10\end{array}$

Hence, the required value of $x$ is $10$ .