Question

In the following exercise, solve for x.

$3{\log }_{3}x={\log }_{3}27$

Short answer

The solution of the given expression is$x=3$

Step-by-step solution

Step 1. Given information. 

The given expression is$3{\log }_{3}x={\log }_{3}27$

Step 2. Use logarithmic properties  

Using power property, we get:

$$\begin{gathered} 3{\log }_{3}x={\log }_{3}27 \\ {\log }_3{x}^3={\log }_{3}27 \end{gathered}$$

Using one to one property, we get:

$x^3=27$

Step 3. Solve for x.   

Solving the equation, we get:

$$\begin{gathered} x^3=27 \\ x=\sqrt[3]{27}\left[cubingrootbothsides\right] \\ x=3 \end{gathered}$$

Step 4. Check the value

Substituting $x=3$in the given expression, we get:

$$\begin{gathered} 3{\log }_{3}x={\log }_{3}27 \\ 3{\log }_{3}3={\log }_{3}27 \\ {\log }_3{3}^3={\log }_{3}27 \\ {\log }_{3}27={\log }_{3}27 \end{gathered}$$

This is true.

Thus $x=3$is the solution of the given function.