Question

In the following exercises, solve each logarithmic equation,${\log }_{a}81=2$

Short answer

The value is$a=9$

Step-by-step solution

Given information

Given the equation

${\log }_{a}81=2$

Rewriting in exponential form

Rewriting, we get

$$\begin{gathered} {\log }_{a}81=2 \\ {a}^2=81 \\ a=\pm 9 \end{gathered}$$

Checking the value

Substituting , we get

$$\begin{gathered} {\log }_9\left(81\right)={\log }_9\left(9^2\right) \\ {\log }_9\left(81\right)=2{\log }_9\left(9\right) \\ {\log }_9\left(81\right)=2 \end{gathered}$$

Base can't be negative so ,$a=-9$is eliminated