Question Number 172082 by Mikenice last updated on 23/Jun/22
$${solve} \\ $$ $${log}_{\mathrm{0}.\mathrm{5}} ^{\mathrm{2}} {x}+{log}_{\mathrm{0}.\mathrm{5}} {x}−\mathrm{2}\underset{−} {<}\mathrm{0} \\ $$
Commented bymokys last updated on 23/Jun/22
$${log}_{\mathrm{0}.\mathrm{5}} ^{\mathrm{2}} {x}+{log}_{\mathrm{0}.\mathrm{5}} {x}\:\leqslant\mathrm{2} \\ $$ $$\mathrm{2}{log}_{\mathrm{0}.\mathrm{5}} {x}+{log}_{\mathrm{0}.\mathrm{5}} {x}\leqslant\mathrm{2} \\ $$ $$\mathrm{3}{log}_{\mathrm{0}.\mathrm{5}} {x}\:\leqslant\mathrm{2}\: \\ $$ $${log}_{\mathrm{0}.\mathrm{5}} {x}\leqslant\frac{\mathrm{2}}{\mathrm{3}} \\ $$ $${x}\leqslant\left(\mathrm{0}.\mathrm{5}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} \:\leqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{4}}}\: \\ $$ $$ \\ $$ $${another}\:{way}\:: \\ $$ $$ \\ $$ $$\frac{{lnx}}{{ln}\left(\mathrm{0}.\mathrm{5}\right)}\leqslant\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:\Rightarrow{lnx}\:\leqslant\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:{ln}\:\left(\mathrm{0}.\mathrm{5}\right) \\ $$ $$ \\ $$ $${lnx}\:\leqslant\:{ln}\left(\mathrm{0}.\mathrm{5}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} \:\Rightarrow\:{x}\:\leqslant\:\left(\mathrm{0}.\mathrm{5}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} \leqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{4}}} \\ $$ $$ \\ $$ $${Aldolaimy} \\ $$