Chapter 4.4, Problem 28E

To determine

To calculate: The expression , ${\log }_2{\left(xy\right)}^{10}$

Answer to Problem 28E

The expended form is, $10\left[{\log }_2\left(x\right)+{\log }_2\left(y\right)\right]$

Explanation of Solution

Given information:

Expend the expression, ${\log }_2{\left(xy\right)}^{10}$

Formula used:

For change the base rule,

  $\begin{array}{l}{\log }_{a}b=\frac{\log b}{\log a},\\ {\log }_a{b}^x=x{\log }_{a}b\end{array}$

Calculation :

Consider the expression, ${\log }_2{\left(xy\right)}^{10}$

Rewrite the expression,

  $\begin{array}{c}{\log }_2{\left(xy\right)}^{10}={\log }_2{\left(x\right)}^{10}+{\log }_2{\left(y\right)}^{10}\\ ={\left[{\log }_2\left(x\right)+{\log }_2\left(y\right)\right]}^{10}\end{array}$

Apply the log base rule: ,

  ${\log }_a\left(x\right)=\frac{{\log }_b\left(x\right)}{{\log }_b\left(a\right)},{\log }_a{b}^x=x{\log }_{a}b$

  ${\left[{\log }_2\left(x\right)+{\log }_2\left(y\right)\right]}^{10}=\frac{1}{10}\left[{\log }_2\left(x\right)+{\log }_2\left(y\right)\right]$

The above expression can’t be simplified further.

Thus the simplified form of the expression is $\frac{1}{10}\left[{\log }_2\left(x\right)+{\log }_2\left(y\right)\right]$