Question

In the following exercise, solve for x.

${\log }_{4}64=2{\log }_{4}x$

Short answer

The solution of the given function is$x=8$

Step-by-step solution

Step 1. Given information

The given information is${\log }_{4}64=2{\log }_{4}x$

Step 2. Use logarithmic properties

Using power property, we get:

$$\begin{gathered} {\log }_{4}64=2{{\log }_4}^x \\ {\log }_{4}64={{\log }_4}^{x^2} \end{gathered}$$

Using one to one property, we get:

$$\begin{gathered} {\log }_{4}64={\log }_4{x}^2 \\ 64=x^2 \end{gathered}$$

Step 3. Solve for x.

Solving the equation, we get:

$$\begin{gathered} x^2=64 \\ x=\pm \sqrt{64}[rootingboothsides] \\ x=\pm 8 \end{gathered}$$

We can eliminate$x=-8$because the logarithmic function doesn't take negative values.

Step 4. Check the values

Substituting $x=8$in the given expression, we get:

$$\begin{gathered} {\log }_{4}64=2{\log }_{4}x \\ {\log }_{4}64=2{\log }_{4}8 \\ {\log }_{4}64={\log }_{4}64 \end{gathered}$$

This is true.

Thus $x=8$is the solution of the given function.