Simplify logarithmic fraction

Numerator = $2 \times 4 \times 8 \times 6 = 384$.

Denominator terms: $\log_2 4 = 2 \implies (\log_2 4)^2 = 4$. $\log_4 8 = \frac{3}{2} \implies (\log_4 8)^3 = \frac{27}{8}$. $\log_8 16 = \frac{4}{3} \implies (\log_8 16)^4 = \frac{256}{81}$.

Denominator = $4 \times \frac{27}{8} \times \frac{256}{81} = 4 \times \frac{1}{3} \times \frac{32}{3} = \frac{128}{3} \times \frac{3}{3} \dots$ Wait: $4 \times \frac{27}{8} \times \frac{256}{81} = \frac{1}{2} \times 27 \times \frac{256}{81} = \frac{256}{6} = 16$.

Fraction = $\frac{384}{16} = 24$.