Chapter 8.5, Problem 4BCYP

To determine

To solve: The value of logarithmic equation ${\log }_{6}x+{\log }_6\left(x+5\right)=2$ .

Answer to Problem 4BCYP

The solution of the equation ${\log }_{6}x+{\log }_6\left(x+5\right)=2$ is $x=4$ .

Explanation of Solution

Given:

The logarithmic equation is ${\log }_{6}x+{\log }_6\left(x+5\right)=2$ .

Calculation:

Simplify the given equation by using Multiplication Rule of logarithms.

  $\begin{array}{c}{\log }_{6}x+{\log }_6\left(x+5\right)=2\\ {\log }_6\left[x\left(x+5\right)\right]=2\\ x\left(x+5\right)=6^2\\ x^2+5x=36\end{array}$

Simplify further.

  $\begin{array}{c}{x}^2+5x-36=0\\ x^2+9x-4x-36=0\\ \left(x+9\right)\left(x-4\right)=0\\ x=4,-9\end{array}$

Take positive value.

Therefore, the solution of the equation ${\log }_{6}x+{\log }_6\left(x+5\right)=2$ is $x=4$ .