Question

Evaluate the above expression​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\rm :\longmapsto\:Solve : \: log_{2}( log_{2}( log_{5}(625) ) )$

$\large\underline{\sf{Solution-}}$

Given

$\rm :\longmapsto\ \: log_{2}( log_{2}( log_{5}(625) ) )$

Let consider,

$\rm :\longmapsto\: log_{5}(625)$

$\rm \: = \: \: log_{5}(5 \times 5 \times 5 \times 5)$

$\rm \: = \: \: log_{5}( {5}^{4} )$

$\rm \: = \: \: 4 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{\blue{\{ \because \: log_{ {a}^{p} }( {a}^{q}) = \dfrac{q}{p} \}}}$

$\boxed{\bf\implies \: log_{5}(625) = 4}$

So,

Given

$\rm :\longmapsto\ \: log_{2}( log_{2}( log_{5}(625) ) )$

$\rm \: = \: \: \: log_{2}( log_{2}( 4 ) )$

$\rm \: = \: \: \: log_{2}( log_{2}( 2 \times 2 ) )$

$\rm \: = \: \: \: log_{2}( log_{2}( {2}^{2} ) )$

$\rm \: = \: \: log_{2}(2) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{\blue{\{ \because \: log_{ {a}^{p} }( {a}^{q}) = \dfrac{q}{p} \}}}$

$\rm \: = \: \: 1 \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{\blue{\{ \because \: log_{ {a}}( {a}) =1\}}}$

Hence,

$\bf\implies \: \: log_{2}( log_{2}( log_{5}(625) ) ) = 1$

Additional Information :-

$\rm :\longmapsto\:logx + logy = logxy$

$\rm :\longmapsto\:logx - logy = log \dfrac{x}{y}$

$\rm :\longmapsto\:log \: {x}^{y} = y \: logx$

$\rm :\longmapsto\: log_{x}(x) = 1$

$\rm :\longmapsto\: log_{x}(y) = \dfrac{logy}{logx}$

$\rm :\longmapsto\:log1 = 0$

$\rm :\longmapsto\: {e}^{ log(x)} = x$

$\rm :\longmapsto\: {e}^{ ylog(x)} = {x}^{y}$

$\rm :\longmapsto\: {x}^{ log_{x}(y) } = y$

$\rm :\longmapsto\: {x}^{z log_{x}(y) } = {y}^{z}$