Question

How do you simplify $\ln {e^2}$?

Answer 1

Hint: To evaluate any logarithmic value first convert them into general form of the logarithmic function and solve. The general form is ${\log _a}b = c \Leftrightarrow {a^c} = b$

Complete step-by-step answer:
The objective of the problem is to evaluate the value of $\ln {e^2}$.
About logarithm:
The logarithm is the inverse function of the exponential. The logarithm simply counts the occurrence of the same factor in repeated multiplication. There are two types of logarithms. They are common logarithm and natural logarithm
Common logarithm: A common logarithm has a fixed base . The fixed base is 10. The common logarithms are denoted as ${\log _{10}}a$ or simply $\log a$. These common logarithms are also called decimal logarithms.
Natural logarithm: The natural logarithm of a number is equal to the power of “e” . where is equal to the 2.718281. it is known as the Napier constant. The natural logarithm is denoted as $A = {e^x}$. Where A is any number.
To evaluate the value first convert the logarithmic function to the general form of the function. The general form is ${\log _a}b = c \Leftrightarrow {a^c} = b$.
Now let us consider \[\ln {e^2} = c\].
Since it is a natural logarithm we can take a value as e.
By definition it only happens when ${e^c} = {e^2}$
Since bases are equal the power must be equal.
Therefore, $c = 2$
Thus, the value of $\ln {e^2}$ is 2.

Note: We can also solve this simply by using the power rule of logarithms. The power rule of logarithms is $\log \left( {{a^b}} \right) = b\log a$. Logarithms are used in solving exponentials of earthquakes, the brightness of stars and so on.