Question

How do you expand \[\log _{b}^{\sqrt{\dfrac{57}{74}}}\]?

Answer 1

Hint: From the question we are asked to find the logarithmic expansion of \[\log _{b}^{\sqrt{\dfrac{57}{74}}}\]. So, for the questions of these kind we will use the basic logarithmic formulae which are \[\log _{c}^{\left( \dfrac{a}{b} \right)}=\log _{c}^{a}-\log _{c}^{b}\] and \[\log _{b}^{{{a}^{n}}}=n\log _{b}^{a}\]. Using the above-mentioned logarithmic formulae, we will simplify the question and get the solution for the required question.

Complete step by step solution:
Firstly, for the question \[\log _{b}^{\sqrt{\dfrac{57}{74}}}\] we will use the basic logarithmic formula which is \[\log _{c}^{\left( \dfrac{a}{b} \right)}=\log _{c}^{a}-\log _{c}^{b}\].
After using the formula, we will simplify the equation. So, the equation will be reduced as follows.
\[\Rightarrow \log _{b}^{\sqrt{\dfrac{57}{74}}}\]
\[\Rightarrow \log _{b}^{\sqrt{\dfrac{57}{74}}}=\log _{b}^{\sqrt{57}}-\log _{b}^{\sqrt{74}}\]
Here after getting the above equation for the further simplification we will use the formulae \[\log _{b}^{{{a}^{n}}}=n\log _{b}^{a}\] to the equation.
Here, Before using the logarithmic formula \[\log _{b}^{{{a}^{n}}}=n\log _{b}^{a}\] in the above equation.
First, we have to write the above equation in that form as \[\log _{b}^{{{a}^{n}}}\]
Now, we will rearrange the equation which we will get in the above form. We will arrange in the above to get the solution to look in a more familiar or an easier way.
So, after rearranging the equation will become as follows.
\[\Rightarrow \log _{b}^{\sqrt{\dfrac{57}{74}}}=\log _{b}^{\sqrt{57}}-\log _{b}^{\sqrt{74}}\]
\[\Rightarrow \log _{b}^{\sqrt{\dfrac{57}{74}}}=\log _{b}^{{{57}^{\dfrac{1}{2}}}}-\log _{b}^{{{74}^{\dfrac{1}{2}}}}\]
Now, we will apply the above formula \[\log _{b}^{{{a}^{n}}}=n\log _{b}^{a}\] for the both the terms in right hand side.
By applying we will get,
\[\Rightarrow \log _{b}^{\sqrt{\dfrac{57}{74}}}=\dfrac{1}{2}\log _{b}^{57}-\dfrac{1}{2}\log _{b}^{74}\]
Therefore, the solution to the given question will be\[ \log _{b}^{\sqrt{\dfrac{57}{74}}}=\dfrac{1}{2}\log _{b}^{57}-\dfrac{1}{2}\log _{b}^{74}\].

Note: Students must have a very good knowledge in the concept of logarithms. Students should recall all the formulas of logarithms while doing this problem. We must know basic formulae like,
\[\log _{c}^{\left( \dfrac{a}{b} \right)}=\log _{c}^{a}-\log _{c}^{b}\],\[\Rightarrow \ln \left( ab \right)=\ln a+\ln b\] and \[\log _{b}^{{{a}^{n}}}=n\log _{b}^{a}\]. Students should not make any calculation mistakes.