Question

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)$

Short answer

a condensed form of the logarithm is${\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)={\log }_5\frac{\left(x^2-16\right)}{{\left(x+4\right)}^2}$

Step-by-step solution

Step 1. Given data 

The given logarithm expression is${\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)$

Step 2. Apply the power properties of the logarithm

Use power properties of the logarithm ${\log }_a\left(m^p\right)=p{\log }_a\left(m\right)$

role="math" localid="1645031061196" ${\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)={\log }_5\left(x^2-16\right)-{\log }_5{\left(x+4\right)}^2$

Step 3. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm ${\log }_a\left(\frac{m}{n}\right)={\log }_a\left(m\right)-{\log }_a\left(n\right)$

localid="1645031141769" $$\begin{gathered} {\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)={\log }_5\left(x^2-16\right)-{\log }_5{\left(x+4\right)}^2 \\ {\log }_5\left(x^2-16\right)-2{\log }_5\left(x+4\right)={\log }_5\frac{\left(x^2-16\right)}{{\left(x+4\right)}^2} \end{gathered}$$