Question

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

${\log }_{\sqrt{3}}\left(9\right)$

Short answer

The exact value of the logarithm is 4.

Step-by-step solution

Step 1. Given information

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

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${\log }_{\sqrt{3}}\left(9\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} {\sqrt{3}}^x=9 \\ {\sqrt{3}}^x={\left(\sqrt{3}\right)}^4 \\ x=4 \\ {\log }_{\sqrt{3}}\left(9\right)=4 \end{gathered}$$

The required value is,

${\log }_{\sqrt{3}}\left(9\right)=4$