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What is the value of $\frac{\left(\log _{27} 9\right)\left(\log _{16} 64\right)}{\log _{4} \sqrt{2}}$ ?
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The correct answer is:
4
Consider $\frac{\left(\log _{27} 9\right)\left(\log _{16} 64\right)}{\log _{4} \sqrt{2}}$
$=\frac{\log _{3^{3}}\left(3^{2}\right) \log _{4^{2}}(4)^{3}}{\log _{2^{2}}\left(2^{1 / 2}\right)}$
$=\frac{\frac{2}{3} \log _{3} 3 \times \frac{3}{2} \log _{4} 4}{\frac{1}{2 \times 2} \log _{2} 2}=\frac{1}{\frac{1}{4}}=4$
$=\frac{\log _{3^{3}}\left(3^{2}\right) \log _{4^{2}}(4)^{3}}{\log _{2^{2}}\left(2^{1 / 2}\right)}$
$=\frac{\frac{2}{3} \log _{3} 3 \times \frac{3}{2} \log _{4} 4}{\frac{1}{2 \times 2} \log _{2} 2}=\frac{1}{\frac{1}{4}}=4$
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