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Question Number 193295 by MATHEMATICSAM last updated on 09/Jun/23

$$\mathbf{Find}\mathbf{the}\mathbf{value}\mathbf{of}\mathit{x}\mathbf{from}\mathbf{the}\mathbf{following}$$$$\mathbf{equations}:$$$$4^{\frac{x}{y}+\frac{y}{x}}=32$$$${\log }_3(x-y)+{\log }_3(x+y)=1$$

Answered by BaliramKumar last updated on 09/Jun/23

$$2$$$$$$

Answered by Frix last updated on 09/Jun/23

$$x=py$$$$4^{p+\frac{1}{p}}=4^{\frac{5}{2}}\Rightarrow p=\frac{1}{2}\vee p=2$$$$x-y>0\Rightarrow p>1\Rightarrow p=2$$$${\log }_3\left(y\right)+{\log }_3\left(3y\right)=1$$$${2\log }_{3}y+1=1\Rightarrow y=1\Rightarrow x=2$$

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