Question Number 193295 by MATHEMATICSAM last updated on 09/Jun/23
$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{following}} \\ $$$$\boldsymbol{\mathrm{equations}}: \\ $$$$\mathrm{4}^{\frac{{x}}{{y}}\:+\:\frac{{y}}{{x}}} \:=\:\mathrm{32} \\ $$$$\mathrm{log}_{\mathrm{3}} \left({x}\:−\:{y}\right)\:+\:\mathrm{log}_{\mathrm{3}} \left({x}\:+\:{y}\right)\:=\:\mathrm{1} \\ $$
Answered by BaliramKumar last updated on 09/Jun/23
$$\mathrm{2} \\ $$$$ \\ $$
Answered by Frix last updated on 09/Jun/23
$${x}={py} \\ $$$$\mathrm{4}^{{p}+\frac{\mathrm{1}}{{p}}} =\mathrm{4}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\Rightarrow\:{p}=\frac{\mathrm{1}}{\mathrm{2}}\vee{p}=\mathrm{2} \\ $$$${x}−{y}>\mathrm{0}\:\Rightarrow\:{p}>\mathrm{1}\:\Rightarrow\:{p}=\mathrm{2} \\ $$$$\mathrm{log}_{\mathrm{3}} \:\left({y}\right)\:+\mathrm{log}_{\mathrm{3}} \:\left(\mathrm{3}{y}\right)\:=\mathrm{1} \\ $$$$\mathrm{2log}_{\mathrm{3}} \:{y}\:+\mathrm{1}=\mathrm{1}\:\Rightarrow\:{y}=\mathrm{1}\:\Rightarrow\:{x}=\mathrm{2} \\ $$