Question

In the following exercises, find the exact value of each logarithm without using a calculator.

${\log }_9\frac{1}{81}$

Short answer

The value of expression${\log }_9\frac{1}{81}=-2.$

Step-by-step solution

Step 1. Given

An expression is given as

${\log }_9\frac{1}{81}$

Step 2. Calculation

Put the expression equal to $x$,

${\log }_9\frac{1}{81}$

Change the equation into exponential form as,

$$\begin{gathered} 9^x=\frac{1}{81} \\ {9}^x=\frac{1}{9^2} \\ {9}^x=9^{-2} \\ comparingthepowersweget, \\ x=-2 \end{gathered}$$