Question

In the following exercises, solve each equation.

${\log }_a\left(64\right)=2$.

Short answer

The solution is$a=8$.

Step-by-step solution

Step 1 Given equation is,

${\log }_a\left(64\right)=2$.

The objective is, we need to find the solution for the given equation.

The standard form of exponential equation is,$x=a^y$.

Step 2 First convert the given equation into the exponential equation,

Here,$y=2,a=a,x=64.$

$$\begin{gathered} 64=a^2 \\ a=\sqrt{64} \\ a=\pm 8 \end{gathered}$$

The logarithm of negative number does not exist.

So, discard role="math" localid="1645008700858" $a=-8$.

So, the solution is,$a=8$.

Step 3 Let us check the answer.

Substitute $a=8$into the given equation.

$$\begin{gathered} {\log }_8\left(64\right)=2 \\ {\log }_8{\left(8\right)}^2=2 \\ 2=2 \end{gathered}$$

which is true.