Question

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

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

Short answer

The result is$3\log x+\frac{1}{2}\log \left(x+1\right)-2\log \left(x-2\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

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

Step 2. Formula used.

The logarithm properties are defined as,

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

Step 3. Calculation.

Apply the logarithm property to simplify.

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