Question

In the following exercise, solve for x.

$3{\log }_{6}x={\log }_{6}64$

Short answer

The solution of the given expression is$x=4$

Step-by-step solution

Step 1. Given information.  

The given expression$3{\log }_{6}x={\log }_{6}64$

Step 2. Use logarithmic properties   

Using power property, we get:

$$\begin{gathered} 3{\log }_{6}x={\log }_{6}64 \\ {\log }_6{x}^3={\log }_{6}64 \end{gathered}$$

Using one to one property, we get:

$x^3=64$

Step 3. Solve for x.    

Solving the equation, we get:

$$\begin{gathered} x^3=64 \\ x=\sqrt[3]{64}\left[cubingrootbothsides\right] \\ x=4 \end{gathered}$$

Step 4. Check the value 

Substituting $x=4$in the given expression, we get:

$$\begin{gathered} 3{\log }_{6}x={\log }_{6}64 \\ 3{\log }_{6}4={\log }_{6}64 \\ {\log }_6{4}^3={\log }_{6}64 \\ {\log }_{6}64={\log }_{6}64 \end{gathered}$$

This is true

Thus $x=4$is the solution of the given function.