$\small \dfrac{2\times4\times8\times16}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4}$ is equal to

$2 \times 4 \times 8 \times 16 = 1024$

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For logarithms like these, we'll use the logarithm property: $\log_a b^n = n \log_a b$

Also, when the base and argument are both powers of the same number, we can use: $\log_{a^m} b^n = \tfrac{n}{m} \log_a b$

Let's calculate each logarithm:

Finding $\log_2 4$:

$\log_2 4 = \log_2 2^2 = 2$

Finding $\log_4 8$:

Since $4 = 2^2$ and $8 = 2^3$:

$\log_4 8 = \log_{2^2} 2^3 = \tfrac{3}{2}$

Finding $\log_8 16$:

Since $8 = 2^3$ and $16 = 2^4$:

$\log_8 16 = \log_{2^3} 2^4 = \tfrac{4}{3}$

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$(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4$

$= (2)^2 \times \left(\tfrac{3}{2}\right)^3 \times \left(\tfrac{4}{3}\right)^4$

$= 4 \times \tfrac{27}{8} \times \tfrac{256}{81}$

$4 \times \tfrac{27}{8} = \tfrac{108}{8} = \tfrac{27}{2}$

$\tfrac{27}{2} \times \tfrac{256}{81} = \tfrac{27 \times 256}{2 \times 81} = \tfrac{6912}{162}$

$\tfrac{6912}{162} = \tfrac{128}{3}$

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$\tfrac{\text{Numerator}}{\text{Denominator}} = \tfrac{1024}{\tfrac{128}{3}} = \tfrac{1024 \times 3}{128} = \tfrac{3072}{128} = 24$

Answer: 24

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When you see logarithms with bases and arguments that are powers of the same number, use the property $\log_{a^m} b^n = \tfrac{n}{m} \log_a b$

This makes complex logarithmic expressions much simpler to handle