Chapter 8, Problem 46SGR

To determine

To solve: The equation $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ and check the solution.

Answer to Problem 46SGR

The exponential for of equation $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ is $x=\sqrt{12}$ .

Explanation of Solution

Given:

The equation is $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ .

Calculation:

Simplify the equation $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ by using Product Rule.

  $\begin{array}{c}2{\log }_{3}x+{\log }_{3}3={\log }_{3}36\\ {\log }_3{x}^2+{\log }_{3}3={\log }_{3}36\\ {\log }_3\left(x^2\cdot 3\right)={\log }_{3}36\\ 3x^2={\log }_{3}36\end{array}$

Simplify further.

  $\begin{array}{c}{x}^2=\frac{36}{3}\\ =12\\ x=\sqrt{12}\end{array}$

Substitute $x=\sqrt{12}$ in the equation $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ to check the solution.

  $\begin{array}{c}2{\log }_3\sqrt{12}+{\log }_{3}3={\log }_{3}36\\ {\log }_3{\left(\sqrt{12}\right)}^2+{\log }_{3}3={\log }_{3}36\\ {\log }_{3}12+{\log }_{3}3={\log }_{3}36\\ {\log }_{3}36={\log }_{3}36\end{array}$

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

Therefore, the solution of the equation $2{\log }_{3}x+{\log }_{3}3={\log }_{3}36$ is $x=\sqrt{12}$ .