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Find-the-value-of-x-from-the-following-equations-4-x-y-y-x-32-log-3-x-y-log-3-x-y-1-




Question Number 193295 by MATHEMATICSAM last updated on 09/Jun/23
Find the value of x from the following  equations:  4^((x/y) + (y/x))  = 32  log_3 (x − y) + log_3 (x + y) = 1
$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{following}} \\ $$$$\boldsymbol{\mathrm{equations}}: \\ $$$$\mathrm{4}^{\frac{{x}}{{y}}\:+\:\frac{{y}}{{x}}} \:=\:\mathrm{32} \\ $$$$\mathrm{log}_{\mathrm{3}} \left({x}\:−\:{y}\right)\:+\:\mathrm{log}_{\mathrm{3}} \left({x}\:+\:{y}\right)\:=\:\mathrm{1} \\ $$
Answered by BaliramKumar last updated on 09/Jun/23
2
$$\mathrm{2} \\ $$$$ \\ $$
Answered by Frix last updated on 09/Jun/23
x=py  4^(p+(1/p)) =4^(5/2)  ⇒ p=(1/2)∨p=2  x−y>0 ⇒ p>1 ⇒ p=2  log_3  (y) +log_3  (3y) =1  2log_3  y +1=1 ⇒ y=1 ⇒ x=2
$${x}={py} \\ $$$$\mathrm{4}^{{p}+\frac{\mathrm{1}}{{p}}} =\mathrm{4}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\Rightarrow\:{p}=\frac{\mathrm{1}}{\mathrm{2}}\vee{p}=\mathrm{2} \\ $$$${x}−{y}>\mathrm{0}\:\Rightarrow\:{p}>\mathrm{1}\:\Rightarrow\:{p}=\mathrm{2} \\ $$$$\mathrm{log}_{\mathrm{3}} \:\left({y}\right)\:+\mathrm{log}_{\mathrm{3}} \:\left(\mathrm{3}{y}\right)\:=\mathrm{1} \\ $$$$\mathrm{2log}_{\mathrm{3}} \:{y}\:+\mathrm{1}=\mathrm{1}\:\Rightarrow\:{y}=\mathrm{1}\:\Rightarrow\:{x}=\mathrm{2} \\ $$

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