Chapter 8.4, Problem 14PPS

To determine

The solution of logarithmic equation.

Answer to Problem 14PPS

  $x=4,-2$ .

Explanation of Solution

Given information:

Solve:

  ${\log }_3(3x+8)={\log }_3(x^2+x)$

Calculation:

The given logarithm inequality is ${\log }_3(3x+8)={\log }_3(x^2+x)$ .

To solve the inequality logarithmic equation by using property of inequality for logarithmic function:

  $\begin{array}{l}{\log }_3(3x+8)={\log }_3(x^2+x)\text{ {The original equation}}\\ (3x+8)=(x^2+x)\text{ {Property of equality for logarithmic function} }\\ -x^2\text{+2}x+8=0\text{ {Subtract }x\text{ and }{x}^2\text{ from each side} }\\ (x-4)(x+2)=0\text{ {Factorisation} }\\ x-4=0\text{ or }x+2=0\text{ {By zero product property} }\\ x=4\text{, or }x=-2\end{array}$

Substitute each value into the original equation:

  $\begin{array}{l}x=4x=-2\\ {\log }_3(3(4)+8)={\log }_3(4^2+4){\log }_3(3(-2)+8)={\log }_3({(}^{-}-2)\\ {\log }_{3}20={\log }_{3}20{\log }_{3}2={\log }_{3}2\end{array}$

Thus, the solution of the equation is $x=4,-2$ .