Question

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

$\ln \left(e^3\right)$

Short answer

The exact value of the logarithm is 3.

Step-by-step solution

Step 1. Given information

A logarithmic function is given as$\ln \left(e^3\right)$.

Step 2. Formula Used

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

Step 3. Finding Value of the logarithm

Consider $\ln \left(e^3\right)=x$

By changing rule of logarithm to exponential function,

$$\begin{gathered} e^x=e^3 \\ {e}^x=e^3 \\ x=3 \\ \ln \left(e^3\right)=3 \end{gathered}$$

The required value is,

$\ln \left(e^3\right)=3$