Question

How do you evaluate \[\ln \dfrac{1}{e}\]?

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.

Complete step by step answer:
We are going to use the formula of logarithm, which is:
\[{\log _x}{x^n} = n\]
The given expression to be simplified is:
\[\ln \dfrac{1}{e}\]
Now, the argument of the function is Euler’s constant.
\[{\log _e}\] is written as \[\ln \]
Hence, we have \[{\log _e}\dfrac{1}{e}\]
From the formula of logarithm,
\[{\log _x}{x^n} = n\]
Now, \[\dfrac{1}{e} = {e^{ - 1}}\]
Hence, \[{\log _e}\dfrac{1}{e} = {\log _e}{e^{ - 1}}\]

Thus, \[\ln \dfrac{1}{e} = - 1\]

Note: The \[\log \] function has other basic properties too:
\[{\log _b}a = n \Rightarrow {b^n} = a\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]
In the given question, we had to solve the given expression containing an expression involving the use of Euler’s constant and the natural log function. The natural log function is a special case of the logarithm function with the base of Euler’s constant. The place where some students get stuck is when they do not know the meaning of the natural log function, and when they do not know the formula of the logarithm functions; without knowing the formula of this concept, it is impossible to solve the question. So, we need to remember the required formula so that we can solve this question.