Question

Prove that log of base 2 [log of base 2 [log of base 2 16]] = 1​

Answer 1

$\huge \boxed{ \red{ \mathfrak{solution}}}$

To prove :

$\bold{log_{2} log_{2} log_{2}(16) = 1}$

LHS

$\leadsto \: log_{2} log_{2} log_{2}(16) \\ \\ \leadsto log_{2} log_{2} log_{2}( {2}^{4} ) \\ \\ \leadsto \: log_{2} log_{2}4 log_{2}(2) \\ \\ \leadsto log_{2} log_{2}(4) \\ \\ \leadsto log_{2} log_{2}( {2}^{2} ) \\ \\ \leadsto \: log_{2}2 log_{2}(2) \\ \\ \leadsto \: log_{2}(2) (1) \\ \\ \leadsto \: (1)(1) \\ \\ \leadsto 1$

LHS=RHS

$\huge \boxed{ \bold{\pink{proved}}}$

Formula used:

$\star{ \bold{ log_{x}(x) = 1}}$

$\star \: \bold{ log_{a}( {b}^{c} ) = c log_{a}(b) }$