Q 29. - Precalculus Enhanced with Graphing Utilities

Question

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

${\log }_{\frac{1}{2}}\left(16\right)$

Step-by-Step Solution

Answer

The exact value of the logarithm is  $-4$.

Step 1. Given information

A logarithmic function is given as ${\log }_{\frac{1}{2}}\left(16\right)$.

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 ${\log }_{\frac{1}{2}}\left(16\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} {\left(\frac{1}{2}\right)}^x=16 \\ {\left(\frac{1}{2}\right)}^x={\left(\frac{1}{2}\right)}^{-4} \\ x=-4 \\ {\log }_{\frac{1}{2}}\left(16\right)=-4 \end{gathered}$$

The required value is,

${\log }_{\frac{1}{2}}\left(16\right)=-4$