Question

How do you simplify \[\ln {e^3}\]?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a function. The function has a constant as its argument. This constant is an irrational number. And this constant is raised to a power. We have to simplify the value of this expression. This can be easily done if we know the relation between the function and the irrational constant. Also, we need to know the property of the function with exponents.

Formula Used:
Here, we are going to use the formula of logarithm, which is:
\[{\log _x}{x^n} = n\]

Complete step-by-step answer:
The given expression to be simplified is:
\[\ln {e^3}\]
Now, the argument of the function is Euler’s constant.
\[{\log _e}\] is written as \[\ln \]
Hence, we have \[{\log _e}{e^3}\]
And, from the formula of logarithm,
\[{\log _x}{x^n} = n\]
We can say that \[\ln {e^3} = 3\]

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _b}a = n \Rightarrow {b^n} = a\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]

Note: In this question, to solve for the answer, we needed to know the properties of the logarithmic function. We needed to know how \[\ln \] and \[\log \] are related. If we know such basic things of any topic, we can easily solve for the answer.