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If $\log_4 5=\left ( \log _{4}y \right )\left ( \log _{6}\sqrt{5} \right )$, then $y$ equals

Given that, $\log_{4}5 = (\log_{4}y) (\log_{6}\sqrt{5})$

$\Rightarrow \dfrac{ \log_{4}5}{ \log_{4}y} = \log_{6} \sqrt{5}$

$\Rightarrow \log_{y}5 = \log_{6} \sqrt{5} \quad \left[ \because \log_{a}b = \frac{\log_{x}b}{\log_{x}a}\right]$

Let $\log_{6} \sqrt{5} = k$

$\Rightarrow \sqrt{5} = 6^{k} \quad [ \because \log_{a}b = x \Rightarrow b = a^{x}]$

On squaring both sides.

$\Rightarrow ( \sqrt{5})^{2} = (6^{k})^{2}$

$\Rightarrow 5 = 6^{2k}$

$\Rightarrow \boxed{36^{k} = 5}$

Now, $\log_{y}5 = k$

$\Rightarrow 5 = y^{k}$

$\Rightarrow y^{k} = 36^{k} \quad [ \because 5 = 36^{k}]$

$\Rightarrow \boxed {y = 36}$

$\textbf{Second Method:}$

Given that, $\log_{4}5 = (\log_{4}y)(\log_{6} \sqrt{5})$

$\Rightarrow \log_{4}y = \dfrac{ \log_{4}5}{\log_{6}5^{\frac{1}{2}}}$

$\Rightarrow \log_{4}y = \dfrac{\log_{4}5}{\frac{1}{2} \log_{6}5} \quad [ \because \log_{b}a^{x} = x \log_{b}a]$

$\Rightarrow \log_{4}y = \dfrac{ 2 \log_{4}5}{\log_{6}5}$

$\Rightarrow \log_{4}y = \dfrac{ 2 \log_{4}5}{\frac{1}{\log_{5}6}} \quad \left[ \because \log_{a}b = \frac{1}{\log_{b}a}\right]$

$\Rightarrow \log_{4}y = 2(\log_{4}5)(\log_{5}6)$

$\Rightarrow \log_{4}y = 2 \left( \frac{\log_{5}6}{\log_{5}4} \right)$

$\Rightarrow \log_{4}y = 2 \log_{4}6$

$\Rightarrow \log_{4}y = \log_{4} 6^{2}$

$\Rightarrow y = 6^{2}$

$\Rightarrow \boxed{y = 36}$

Correct Answer$: 36$

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