Question

Find the value of ${\log }_5{\log }_2{\log }_3{\log }_{2}512$

Answer 1

Let $y={\log }_5{\log }_2{\log }_3{\log }_{2}512$

$\Rightarrow y={\log }_5{\log }_2{\log }_3{\log }_2{2}^9$

$\Rightarrow y={\log }_5{\log }_2{\log }_3\left(9{\log }_{2}2\right)$

$\Rightarrow y={\log }_5{\log }_2{\log }_3{3}^2$ (${\log }_{a}a=1$ if $a>1a\ne 1$)

$\Rightarrow y={\log }_5{\log }_{2}2={\log }_{5}1=0$
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