Question

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

$\ln \left(\sqrt{e}\right)$

Short answer

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

Step-by-step solution

Step 1. Given information

A logarithmic function is given as$\ln \left(\sqrt{e}\right)$.

Step 2. Formula Used

The logarithmic function $y=\ln x$ can be changed into an exponential statement as $x=e^y$.

Step 3. Finding Value of the logarithm 

Consider $\ln \left(\sqrt{e}\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} e^x=\sqrt{e} \\ {e}^x=e^{\frac{1}{2}} \\ x=\frac{1}{2} \\ \ln \left(\sqrt{e}\right)=\frac{1}{2} \end{gathered}$$

The required value is,

$\ln \left(\sqrt{e}\right)=\frac{1}{2}$