Calculation:

\(\frac{2\times 4 \times 8 \times 16}{(log_24)^2 (log_48)^2(log _816)^4}\)

⇒ \(\frac{{2{\rm{\;}} \times 4 \times 8 \times 16}}{{{{({\rm{lo}}{{\rm{g}}_2}4{\rm{\;}} \times {\rm{\;lo}}{{\rm{g}}_4}8{\rm{\;}} \times {\rm{lo}}{{\rm{g}}_8}16)}^2}{\rm{\;}} \times \left( {{\rm{lo}}{{\rm{g}}_4}8 \times {\rm{lo}}{{\rm{g}}_8}16} \right) \times {\rm{lo}}{{\rm{g}}_8}16}}\)

We know \(lo{g_{b\;}}a = \frac{{loga}}{{logb}}\)

⇒ \(\frac{{\left( {2 \times 4 \times 8 \times 16} \right)}}{{{{(\frac{{{\bf{log}}4}}{{{\bf{log}}2}} \times \frac{{{\bf{log}}8}}{{{\bf{log}}4}} \times \frac{{{\bf{log}}16}}{{{\bf{log}}8}})}^{2\;}} \times \left( {\frac{{{\bf{log}}8}}{{{\bf{log}}4}} \times \frac{{{\bf{log}}16}}{{{\bf{log}}8}}} \right) \times {\rm{lo}}{{\rm{g}}_8}16}}\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{(lo{g_2}16)}^2} \times \;{\rm{lo}}{{\rm{g}}_4}16 \times {\rm{lo}}{{\rm{g}}_8}16}}\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{(lo{g_2}{2^4})}^2} \times \;{\rm{lo}}{{\rm{g}}_4}{4^2} \times {\rm{lo}}{{\rm{g}}_8}\left( {8 \times 2} \right)}}\)

We know \(lo{g_{a\;}}a = 1\;\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{\left( 4 \right)}^2} \times 2 \times {\rm{lo}}{{\rm{g}}_8}\left( {8 \times 2} \right)}}\)

We know  \(\log \left( {m\; \times n} \right)\; = logm + logn\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{\left( 4 \right)}^2} \times 2 \times \left[ {{\rm{lo}}{{\rm{g}}_8}8 + {\rm{lo}}{{\rm{g}}_8}2} \right]}}\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{\left( 4 \right)}^2} \times 2 \times \left[ {1 + \frac{{{\rm{log}}2}}{{{\rm{log}}8}}} \right]}}\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{\left( 4 \right)}^2} \times 2 \times \left[ {1 + \frac{{{\rm{log}}2}}{{{\rm{log}}{2^3}}}} \right]}}\)

⇒ \(\frac{{2\; \times 4 \times 8 \times 16}}{{{{\left( 4 \right)}^2} \times 2 \times \left[ {1 + \frac{1}{3}} \right]}}\)

⇒ 24

∴ \(\frac{2\times 4 \times 8 \times 16}{(log_24)^2 (log_48)^2(log _816)^4}\) is 24.