Chapter 5.5, Problem 51AYU

To determine

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

Answer to Problem 51AYU

  $\boxed{\log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)=\log x+\log \left(x+2\right)-2\log \left(x+3\right)}$

Explanation of Solution

Given information:

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

  $\log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)x>0$

Calculation:

Consider the given expression,

  $\log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)x>0$

Apply logarithm property.

  ${\log }_a\left(\frac{M}{N}\right)={\log }_{a}M-{\log }_{a}N$

  $\log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)=\log x\left(x+2\right)-\log {\left(x+3\right)}^2$

Now apply, ${\log }_a\left(MN\right)={\log }_{a}M+{\log }_{a}N$ and ${\log }_a{M}^r=r{\log }_{a}M$ property.

  $\begin{array}{l}=\log x+\log \left(x+2\right)-\log {\left(x+3\right)}^2\\ \log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)=\log x+\log \left(x+2\right)-2\log \left(x+3\right)\end{array}$

Hence, $\boxed{\log \left(\frac{x\left(x+2\right)}{{\left(x+3\right)}^2}\right)=\log x+\log \left(x+2\right)-2\log \left(x+3\right)}$