Chapter 4.6, Problem 38E

Solution

To calculate: The value of $x$.

  $-8$ , $-2$

Given:

An equation ${\log }_4\left(-x\right)+{\log }_4\left(x+10\right)=2$ is given.

Concept used:

Product property of logarithm; ${\log }_b\left(mn\right)={\log }_{b}m+\log n$ , $b$ is the base of logarithm and $b$ is also a positive number.

Calculation:

  ${\log }_4\left(-x\right)+{\log }_4\left(x+10\right)=2$

By the product property of logarithm.

  ${\log }_4\left(-x^2-10x\right)=2$

Exponentiation each side by 2

  $\begin{array}{c}{4}^{\log \left(-x^2-10x\right)}=4^2\left[b^{{\log }_{b}x}=x\right]\\ -x^2-10x=16\\ -x^2-10x-16=0\\ x^2+10x+16=0\end{array}$

Thus;

  $\begin{array}{c}{x}^2+8x+2x+16=0\\ x\left(x+8\right)+2\left(x+8\right)=0\\ \left(x+8\right)\left(x+2\right)=0\end{array}$

Thus, $x=-8$ , $x=-2$

Conclusion:

Hence by the property of logarithm the value of $x$ is $-2$ and $-8$.