Question

How do you simplify $\ln \left( {\dfrac{1}{e}} \right)$?

Answer 1

Here we will use the formula of the natural logarithm and get the value which is required of the function $\ln \left( {\dfrac{1}{e}} \right)$ and the formula we can use will be $\ln \left( {\dfrac{a}{b}} \right) = \ln a - \ln b$ and we must know that $\ln$ represents the natural log and its base is always $e$ and also that the log of $1 = 0$.

Complete step-by-step answer:
Here we are given to find the value of $\ln \left( {\dfrac{1}{e}} \right)$
So to solve it, we must know that $\ln$ represents the natural log and its base is always $e$ and also that the log of $1 = 0$
Also we must know that property of logarithm that ${\log _a}a = 1$ $- - - - (1)$
Which means that when base and the number whose log is to be found are equal, then the value of it is always equal to $1$
So we can write the function
$\ln \left( {\dfrac{1}{e}} \right)$ $= \ln 1 - \ln e$
As we know that $\ln \left( {\dfrac{a}{b}} \right) = \ln a - \ln b$
Also we know that $\ln$ represents the natural log and its base is always $e$ and also that the log of $1 = 0$
So we can say that
$\ln \left( {\dfrac{1}{e}} \right)$ $= \ln 1 - \ln e$
$= 0 - {\log _e}e$
So according to the equation (1) we get that:
$\ln \left( {\dfrac{1}{e}} \right)$ $= \ln 1 - \ln e$ $= 0 - 1 = - 1$
So we get that the value of the given function which is $\ln \left( {\dfrac{1}{e}} \right)$ as $- 1$
Hence we can say that in order to solve such problems which contain the logarithm, we must know the properties of the logarithm and the natural log.
So we can say that properties are to be known to solve such problems.

Note: In order to solve such problems where we have to simplify the properties of the logarithm and the natural log must be known to us like:
$\ln \left( {\dfrac{a}{b}} \right) = \ln a - \ln b$
$\ln \left( {ab} \right) = \ln a + \ln b$