Q 30. - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value of the logarithm without using a calculator.

${\log }_{\frac{1}{3}}\left(9\right)$.

Step-by-Step Solution

Answer

The exact value of the logarithm is $-2$.

Step 1. Given information

A logarithmic function is given as ${\log }_{\frac{1}{3}}\left(9\right)$.

Step 2. Formula Used

The logarithmic function $y={\log }_a\left(x\right)$ can be changed into an exponential statement as $x=a^y$.

Step 3. Finding Value of the logarithm

Consider ${\log }_{\frac{1}{3}}\left(9\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} {\left(\frac{1}{3}\right)}^x=9 \\ {\left(\frac{1}{3}\right)}^x=3^2 \\ {\left(\frac{1}{3}\right)}^x={\left(\frac{1}{3}\right)}^{-2} \\ x=-2 \\ {\log }_{\frac{1}{3}}\left(9\right)=-2 \end{gathered}$$

The required value is,

${\log }_{\frac{1}{3}}\left(9\right)=-2$