Question Number 151013 by mathdanisur last updated on 17/Aug/21
$$\mathrm{If}\:\:\mathrm{log}\left(\mathrm{log}\boldsymbol{\mathrm{x}}\right)^{\frac{\mathrm{3}\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{logx}}\right)}{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{logx}}\right)\right)}\:} \:=\:\mathrm{27} \\ $$$$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$
Answered by Olaf_Thorendsen last updated on 17/Aug/21
$$\mathrm{log}\left(\mathrm{log}{x}\right)^{\frac{\mathrm{3log}\left(\mathrm{log}{x}\right)}{\mathrm{log}\left(\mathrm{log}\left(\mathrm{log}{x}\right)\right)}} \:=\:\mathrm{27}\:=\:\mathrm{3}^{\mathrm{3}} \\ $$$$\frac{\mathrm{3log}\left(\mathrm{log}{x}\right)}{\mathrm{log}\left(\mathrm{log}\left(\mathrm{log}{x}\right)\right)}\mathrm{log}\left(\mathrm{log}\left(\mathrm{log}{x}\right)\right)\:=\:\mathrm{3log3} \\ $$$$\mathrm{3log}\left(\mathrm{log}{x}\right)\:=\:\mathrm{3log3} \\ $$$$\mathrm{log}{x}\:=\:\mathrm{3} \\ $$$${x}\:=\:{e}^{\mathrm{3}} \\ $$
Commented by mathdanisur last updated on 17/Aug/21
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$
Commented by Olaf_Thorendsen last updated on 17/Aug/21
$$ \\ $$$$\mathrm{In}\:\mathrm{my}\:\mathrm{country}\:\mathrm{we}\:\mathrm{write} \\ $$$$\mathrm{ln}{e}^{{x}} \:=\:{x}\:\mathrm{and}\:\mathrm{log10}^{{x}} \:=\:{x} \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{in}\:\mathrm{some}\:\mathrm{other}\:\mathrm{countries}\:\mathrm{they}\:\mathrm{write} \\ $$$$\mathrm{log}{e}^{{x}} \:=\:{x}\:\mathrm{and}\:\mathrm{Log10}^{{x}} \:=\:{x} \\ $$