Q 31. - Precalculus Enhanced with Graphing Utilities

Question

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

${\log }_{10}\left(\sqrt{10}\right)$

Step-by-Step Solution

Answer

The exact value of the logarithm is  $\frac{1}{2}$.

Step 1. Given information

A logarithmic function is given as ${\log }_{10}\left(\sqrt{10}\right)$.

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 ${\log }_{10}\left(\sqrt{10}\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} {10}^x=\sqrt{10} \\ {10}^x={10}^{\frac{1}{2}} \\ x=\frac{1}{2} \\ {\log }_{10}\left(\sqrt{10}\right)=\frac{1}{2} \end{gathered}$$

The required value is,

${\log }_{10}\left(\sqrt{10}\right)=\frac{1}{2}$