Chapter 8.5, Problem 9CYU

To determine

To solve: The value of logarithmic equation ${\log }_{3}2x+{\log }_{3}7={\log }_{3}28$ and check the solution.

Answer to Problem 9CYU

The solution of the equation ${\log }_{3}2x+{\log }_{3}7={\log }_{3}28$ is $x=2$ .

Explanation of Solution

Given:

The logarithmic equation is ${\log }_{3}2x+{\log }_{3}7={\log }_{3}28$ .

Calculation:

Simplify the given equation by using Product Rule of logarithms.

  $\begin{array}{c}{\log }_{3}2x+{\log }_{3}7={\log }_{3}28\\ {\log }_3\left(2x\cdot 7\right)={\log }_{3}28\\ {\log }_{3}14x={\log }_{3}28\end{array}$

Use the property if ${\log }_{b}x={\log }_{b}y$ then $x=y$ to solve the equation ${\log }_{3}14x={\log }_{3}28$ .

  $\begin{array}{c}14x=28\\ x=\frac{28}{14}\\ =2\end{array}$

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

  $\begin{array}{c}{\log }_{3}2\left(2\right)+{\log }_{3}7={\log }_{3}28\\ {\log }_{3}4+{\log }_{3}7={\log }_{3}28\\ {\log }_3\left(4\cdot 7\right)={\log }_{3}28\\ {\log }_{3}28={\log }_{3}28\end{array}$

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

Therefore, the solution of the equation ${\log }_{3}2x+{\log }_{3}7={\log }_{3}28$ is $x=2$ .