Question

In the following exercises,solve each logarithmic equation.

${\log }_{a}32=2$.

Short answer

$4\sqrt{2}$

Step-by-step solution

Step 1.Given information

Consider the logarithmic equation ${\log }_{a}32=2$.

The logarithmic from$y={\log }_{a}x$ is equivalent to exponential form$x=a^y$.

Step 2.Solve the equation and find the value of a.

Comparing ${\log }_{a}32=2$with the standard form $y={\log }_{a}x$,you have

$$\begin{gathered} y=2 \\ a=a \\ x=32 \end{gathered}$$

Convert logarithmic equation in exponential equation as shown below.

$$\begin{gathered} x=a^y \\ 32=a^2 \\ {a}^2=32 \end{gathered}$$

So,the equation in the exponential form is $a^2=32$

Now solve the equation using square root property

$$\begin{gathered} a=\pm \sqrt{32} \\ =\pm 4\sqrt{2} \\ Becausethebasecan\text{'}tbenegtive,soeliminatea=-4\sqrt{2} \\ So,thesolutionisa=4\sqrt{2} \end{gathered}$$

Step 3.Check

To check the solution,replace a by $4\sqrt{2}$ in original equation.

$$\begin{gathered} {\log }_{a}32=2 \\ {\log }_{4\sqrt{2}}32=2[a=4\sqrt{2]} \\ {\left(4\sqrt{2}\right)}^2=32 \\ 32=32[true] \\ Therefore,thesolutionisa=4\sqrt{2} \end{gathered}$$