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Question Number 98885 by  M±th+et+s last updated on 16/Jun/20
find the range    f(x)=log_4 log_2 log_(1/2) (x)
$${find}\:{the}\:{range} \\ $$$$ \\ $$$${f}\left({x}\right)={log}_{\mathrm{4}} {log}_{\mathrm{2}} {log}_{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right) \\ $$
Commented by MJS last updated on 17/Jun/20
f(x) defined for 0<x<(1/2) ⇒ −∞<f(x)<+∞  log_(1/2)  x =−log_2  x  ⇒ x>0  log_2  (−log_2  x)  ⇒ −log_2  x >0 ⇒ 0<x<1  log_4  log_2  (−log_2  x)  ⇒ log_2  (−log_2  x) >0 ⇒ 0<x<(1/2)
$${f}\left({x}\right)\:\mathrm{defined}\:\mathrm{for}\:\mathrm{0}<{x}<\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:−\infty<{f}\left({x}\right)<+\infty \\ $$$$\mathrm{log}_{\mathrm{1}/\mathrm{2}} \:{x}\:=−\mathrm{log}_{\mathrm{2}} \:{x} \\ $$$$\Rightarrow\:{x}>\mathrm{0} \\ $$$$\mathrm{log}_{\mathrm{2}} \:\left(−\mathrm{log}_{\mathrm{2}} \:{x}\right) \\ $$$$\Rightarrow\:−\mathrm{log}_{\mathrm{2}} \:{x}\:>\mathrm{0}\:\Rightarrow\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\mathrm{log}_{\mathrm{4}} \:\mathrm{log}_{\mathrm{2}} \:\left(−\mathrm{log}_{\mathrm{2}} \:{x}\right) \\ $$$$\Rightarrow\:\mathrm{log}_{\mathrm{2}} \:\left(−\mathrm{log}_{\mathrm{2}} \:{x}\right)\:>\mathrm{0}\:\Rightarrow\:\mathrm{0}<{x}<\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by  M±th+et+s last updated on 17/Jun/20
thank you sir
$${thank}\:{you}\:{sir} \\ $$

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