Question

How do you evaluate $\ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right)?$

Answer 1

Hint: Here we have to simplify $\ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right)$. Note that $\ln $ represents logarithmic function and it is a natural logarithm. By using the properties of logarithm we try to simplify the given function.
We make use of the following properties of logarithm to solve the above problem.
(1) $\ln \left( {\dfrac{x}{y}} \right) = \ln x - \ln y$
(2) $\ln {x^n} = n\ln x$
(3) ${\log _e}e = 1$
Note that here $e$ is called an exponential constant.

Complete step by step solution:
Here we are given a function $\ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right)$.
The given function is a logarithmic function. The logarithmic function is represented as ${\log _b}a$, where b is called the base and a is a number.
Here we have given the natural logarithmic function where it’s base is $e$ and it is represented as $\ln $.
Now we make use of some basic properties of logarithmic function to evaluate it.
Note that the given function is of the form of $\ln \left( {\dfrac{x}{y}} \right)$.
We have the property related to it and it is given as $\ln \left( {\dfrac{x}{y}} \right) = \ln x - \ln y$.
Note that here $x = 1$ and $y = {\sqrt e ^7}$
Now applying the property, we have,
$\ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = \ln 1 - \ln {\sqrt e ^7}$
Note that $\ln 1 = 0$ and we can write ${\sqrt e ^7} = {e^{\dfrac{7}{2}}}$.
On substituting this we get,
$ \Rightarrow \ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = 0 - \ln {e^{\dfrac{7}{2}}}$
$ \Rightarrow \ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = - \ln {e^{\dfrac{7}{2}}}$
Now this is of the form $\ln {x^n}$. We have the property related to it and it is given by
$\ln {x^n} = n\ln x$.
Note that here $n = \dfrac{7}{2}$.
Now applying the property we get,
$ \Rightarrow \ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = - \dfrac{7}{2}\ln e$
Note that $\ln e = {\log _e}e = 1$
Hence, we get it as,
$ \Rightarrow \ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = - \dfrac{7}{2} \cdot 1$
$ \Rightarrow \ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = - \dfrac{7}{2}$

Hence we get $\ln \left( {\dfrac{1}{{{{\sqrt e }^7}}}} \right) = - \dfrac{7}{2}$.

Note:
If the question has the word log or $\ln $, it represents the given function as a logarithmic function. Note that we have two types of logarithmic function.
One is a common logarithmic function which is represented as a log and its base is 10.
The other one is a natural logarithmic function represented as $\ln $ and it’s base is $e$.
We must know the basic properties of logarithmic functions and note that these properties hold for both log and $\ln $ functions.
Here we must be careful while applying the property related to logarithmic function.
Some properties of logarithmic functions are given below.
(1)$\ln (x \cdot y) = \ln x + \ln y$
(2) $\ln \left( {\dfrac{x}{y}} \right) = \ln x - \ln y$
(3) $\ln {x^n} = n\ln x$
(4) $\ln 1 = 0$
(5) ${\log _e}e = 1$