Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors.

${\log }_5\left(\frac{\sqrt[3]{x^2+1}}{x^2-1}\right)$

Short answer

The result is$\begin{array}{rcl}& & \frac{1}{3}{\log }_5\left(x^2+1\right)+{\log }_5\left(x+1\right)+{\log }_5\left(x-1\right)\end{array}$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

${\log }_5\left(\frac{\sqrt[3]{x^2+1}}{x^2-1}\right)$

Step 2. Formula used.

The logarithm properties are defined as,

$\begin{array}{l}{\log }_a\left(\frac{M}{N}\right)={\log }_{a}M-{\log }_{a}N\\ {\log }_a{M}^r=r{\log }_{a}M\end{array}$

Step 3. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{rcl}{\log }_5\left(\frac{\sqrt[3]{x^2+1}}{x^2-1}\right)& =& {\log }_5\left(\sqrt[3]{x^2+1}\right)+{\log }_5\left(x^2-1\right)\\ & =& {\log }_5{\left(x^2+1\right)}^{\frac{1}{3}}+{\log }_5\left(x+1\right)\left(x-1\right)\\ & =& \frac{1}{3}{\log }_5\left(x^2+1\right)+{\log }_5\left(x+1\right)+{\log }_5\left(x-1\right)\end{array}$