Question

\(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4}\) equals

Answer 1

\(log_24=2;log_48=\frac{3}{2};log_816=\frac{4}{3}\)

\(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4}\)

= \(\frac{2×4×8×16}{(2)^2×\bigg(\frac{3}{2}\bigg)^3×\bigg(\frac{4}{3}\bigg)^4} = 24\)

Answer 2

We have, \(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4}\)
Now we have denominator \(\log_{2} 4 = 2\)
\(\log_{4}{8} = \frac{\log_8}{\log_4} = \frac{\log_{2}{8}}{\log_{2}{4}} = \frac{3}{2}\)

and,\(\log_{8}16 = \frac{\log_{16}}{\log_{8}} = \frac{\log_{2}{16}}{\log_{2}{8}} = \frac{4}{3}\)

\(⇒\)Denominator= \(2 \times 2 \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3}\)
Thus, the fraction turns into \(\frac{{2 \times 4 \times 8 \times 16 \times 3}}{{2 \times 4 \times 4 \times 4}}\)
\(= 2 × 4 × 3 = 24\)