Question

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

$\frac{1}{6}\log x-3\log (x+5)$

Short answer

The simplified expression is $\log \frac{\sqrt[6]{x}}{(x+5{)}^3}$.

Step-by-step solution

Step 1. Given Information 

We are given an expression $\frac{1}{6}\log x-3\log (x+5)$.

Step 2. Solving the expression

Using power property of logarithm, we get

$\frac{1}{6}\log x-3\log (x+5)=\log x^{\frac{1}{6}}-\log (x+5{)}^3$

Using quotients property of logarithm, we get

localid="1645095131607" $$\begin{gathered} \frac{1}{6}\log x-3\log (x+5)=\log \frac{x^{\frac{1}{6}}}{(x+5{)}^3} \\ \frac{1}{6}\log x-3\log (x+5)=\log \frac{\sqrt[6]{x}}{(x+5{)}^3} \end{gathered}$$