Question

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

Answer 1

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.