Chapter 8.3, Problem 25PPS

To determine

To Evaluate: The expression.

Answer to Problem 25PPS

  ${\log }_3\frac{1}{9}=-2$

Explanation of Solution

Given: ${\log }_3\frac{1}{9}$

Formula used:

  $\begin{array}{l}{\log }_{b}x=yifandonlyif\\ b^y=x\end{array}$

Given: ${\log }_3\frac{1}{9}$

Let the expression is equals to $y$

Then

Using the logarithm form

  ${\log }_{b}x=yif{b}^y=x$

  $\begin{array}{l}{\log }_3\frac{1}{9}=y\\ b=3,x=\frac{1}{9}\\ then\\ 3^y=\frac{1}{9}\\ 3^y=3^{-2}\\ y=-2\\ {\log }_3\frac{1}{9}=-2\end{array}$

Hence, ${\log }_3\frac{1}{9}=-2$