Chapter 8.5, Problem 10CYU

To determine

To solve: The value of logarithmic equation $3{\log }_{2}x={\log }_{2}8$ and check the solution.

Answer to Problem 10CYU

The solution of the equation $3{\log }_{2}x={\log }_{2}8$ is $x=2$ .

Explanation of Solution

Given:

The logarithmic equation is $3{\log }_{2}x={\log }_{2}8$ .

Calculation:

Simplify the given equation by using Power Rule of logarithms.

  $\begin{array}{c}3{\log }_{2}x={\log }_{2}8\\ {\log }_2{x}^3={\log }_{2}8\end{array}$

Use the property if ${\log }_{b}x={\log }_{b}y$ then $x=y$ to solve the equation ${\log }_2{x}^3={\log }_{2}8$ .

  $\begin{array}{c}{x}^3=8\\ x=\sqrt[3]{8}\\ =2\end{array}$

Substitute $x=2$ in the given equation to check the solution.

  $\begin{array}{c}3{\log }_{2}2={\log }_{2}8\\ {\log }_2{2}^3={\log }_{2}8\\ {\log }_{2}8={\log }_{2}8\end{array}$

The left hand side is equal to the right hand side. So, the solution is correct.

Therefore, the solution of the equation $3{\log }_{2}x={\log }_{2}8$ is $x=2$ .