Chapter 8.5, Problem 25PPS

To determine

To find: The solution of given expression.

Answer to Problem 25PPS

The value of $x$ is $2$ .

Explanation of Solution

Given:

The given expression is:

  $5{\log }_2\left(x\right)={\log }_{2}32$

The solution is calculated as:

  $\begin{array}{c}5{\log }_2\left(x\right)={\log }_2\left(32\right)\\ {\log }_2\left(x^5\right)={\log }_2\left(32\right)\left[\because {\log }_p\left(a^n\right)=n{\log }_p\left(a\right)\right]\\ x^5=32\left[{\log }_{b}x={\log }_{b}y\to x=y\right]\\ x=2\end{array}$

Check:

  $\begin{array}{c}{\log }_2\left(4\cdot 2\right)+{\log }_2\left(5\right)={\log }_2\left(40\right)\\ {\log }_2\left(8\cdot 5\right)={\log }_2\left(40\right)\\ {\log }_{2}40={\log }_{2}40\left[\text{true}\right]\end{array}$

Therefore the value of $x$ is $2$ .