Question

IF log² (log³(log²x))=1 show that x=512​

Answer 1

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

$\sf \: log_{2}( log_{3}( log_{2}(x) ) ) = 1, \: show \: that \: x \: = \: 512$

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

We know that,

$\boxed{ \sf \: If \: log_{a}(b) = c \: then \: b \: = \: {a}^{c} }$

So,

Consider,

$\rm :\longmapsto\:\sf \: log_{2}( log_{3}( log_{2}(x) ) ) = 1$

$\rm :\implies\: log_{3}( log_{2}(x) ) = {2}^{1}$

$\rm :\implies\: log_{3}( log_{2}(x) ) = {2}$

$\rm :\implies\: log_{2}(x) = {3}^{2}$

$\rm :\longmapsto\: log_{2}(x) = 9$

$\rm :\implies\:x = {2}^{9}$

$\rm :\longmapsto\:x = 512$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

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Additional Information :-

$\rm :\longmapsto\: log(x) + log(y) = log(xy)$

$\rm :\longmapsto\: log(x) - log(y) = log(\dfrac{x}{y} )$

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

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

$\rm :\longmapsto\: log(1) = 0$