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$\dfrac{2 \times 4 \times 8 \times 16} {(\log_{2} 4)^{2} (\log_{4} 8)^{3} (\log_{8} 16)^{4}}$ equals

Given that, $\dfrac{2 \times 4 \times 8 \times 16}{ \left( \log_{2}4 \right)^{2} \left( \log_{4}8 \right)^{3} \left( \log_{8}16 \right)^{4}}$

$\Rightarrow \dfrac{2 \times 4 \times 8 \times 16}{ \left( \log_{2}2^{2} \right)^{2} \left( \log_{2^{2}}2^{3} \right)^{3} \left( \log_{2^{3}}2^{4} \right)^{4}}$

$\Rightarrow \dfrac{2 \times 4 \times 8 \times 16}{ \left(2 \log_{2}2 \right)^{2} \left( \frac{3}{2} \log_{2}2 \right)^{3} \left( \frac{4}{3} \log_{2}2 \right)^{4}} \quad [\because \log_{a^{m}} x^{b} = \frac{b}{m} \log _{a} x]$

$\Rightarrow \dfrac{2 \times 4 \times 8 \times 16}{2^{2} \times \left( \frac{3}{2} \right)^{3} \times \left( \frac{4}{3} \right)^{4}} \quad [\because \log_{a} a = 1]$

$\Rightarrow \dfrac{2 \times 4 \times 8 \times 16}{4 \times \frac{3^{3}} {8} \times \frac{4 \times 4 \times 4 \times 4}{3 \times 3^{3}}}$

$\Rightarrow 24$

Correct Answer: $24$
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