Chapter 8.4, Problem 15PPS

To determine

The solution of logarithmic equation.

Answer to Problem 15PPS

  $x=4,-3$ .

Explanation of Solution

Given information:

Solve:

  ${\log }_{12}(x^2-7)={\log }_{12}(x+5)$

Calculation:

The given logarithm inequality is ${\log }_{12}(x^2-7)={\log }_{12}(x+5)$ .

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

  $\begin{array}{l}{\log }_{12}(x^2-7)={\log }_{12}(x+5)\text{ {The original equation}}\\ (x^2-7)=(x+5)\text{ {Property of equality for logarithmic function} }\\ x^2-x-12=0\text{ {Subtract }x\text{ and 5 from each side} }\\ (x-4)(x+3)=0\text{ {Factorisation} }\\ x-4=0\text{ or }x+3=0\text{ {By zero product property} }\\ x=4\text{, or }x=-3\end{array}$

Substitute each value into the original equation:

  $\begin{array}{l}x=4x=-3\\ {\log }_{12}(4^2-7)={\log }_{12}(4+5){\log }_{12}({(}^{-}-7)={\log }_{12}((-3+5)\\ {\log }_{12}9={\log }_{12}\text{9 }{\log }_{12}2={\log }_{12}2\end{array}$

Thus, the solution of the equation is $x=4,-3$ .