Q 33. - Precalculus Enhanced with Graphing Utilities

Question

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

${\log }_{\sqrt{2}}\left(4\right)$

Step-by-Step Solution

Answer

The exact value of the logarithm is 4.

Step 1. Given information

A logarithmic function is given as ${\log }_{\sqrt{2}}\left(4\right)$.

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 ${\log }_{\sqrt{2}}\left(4\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} {\sqrt{2}}^x=4 \\ {\sqrt{2}}^x={\sqrt{2}}^4 \\ x=4 \\ {\log }_{\sqrt{2}}\left(4\right)=4 \end{gathered}$$

The required value is,

${\log }_{\sqrt{2}}\left(4\right)=4$