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  4. If log 2[ log 3 log4( log 5x) ]=0, then the value of x is

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Q. If $ {{\log }_{2}}[{{\log }_{3}}\{lo{{g}_{4}}({{\log }_{5}}x)\}]=0, $ then the value of x is

J & K CETJ & K CET 2003

Solution:

Given that, $ {{\log }_{2}}[{{\log }_{3}}\{{{\log }_{4}}({{\log }_{5}}x)\}]=0 $
$ \Rightarrow $ $ {{\log }_{3}}\{{{\log }_{4}}({{\log }_{5}}x)\}={{2}^{0}}=1 $
$ \Rightarrow $ $ {{\log }_{4}}({{\log }_{5}}x)={{3}^{1}}=a $
$ \Rightarrow $ $ {{\log }_{5}}\,\,x={{4}^{3}}=64 $
$ \Rightarrow $ $ x={{5}^{64}} $