Question

If log{{x+y}/3}=1/2{logx+log Y}, then find the value of x/y+y/x?

Answer 1

$log( \frac{x+y}{3} )= \frac{1}{2} (logx+logy)\\ \\ \Rightarrow log( \frac{x+y}{3} )= \frac{1}{2} (logxy)=log(xy)^{ \frac{1}{2} }=log( \sqrt{xy}) \\ \\ \Rightarrow \frac{x+y}{3} = \sqrt{xy} \\ \\ \Rightarrow x+y = 3\sqrt{xy} \\ \\ \Rightarrow \frac{x+y}{\sqrt{xy} } = 3\\ \\ \Rightarrow \frac{x}{\sqrt{xy} }+ \frac{y}{\sqrt{xy} } = 3\\ \\ \Rightarrow \frac{ \sqrt{x} }{\sqrt{y} }+ \frac{ \sqrt{y} }{\sqrt{x} } = 3$

Squaring both sides

$\\ \\ \Rightarrow (\frac{ \sqrt{x} }{\sqrt{y} }+ \frac{ \sqrt{y} }{\sqrt{x} })^2 = 3^2\\ \\ \Rightarrow (\frac{ \sqrt{x} }{\sqrt{y} })^2+ (\frac{ \sqrt{y} }{\sqrt{x} })^2+2 \times \frac{ \sqrt{x} }{\sqrt{y} } \times \frac{ \sqrt{x} }{\sqrt{y} } = 3^2 \\ \\ \Rightarrow \frac{x}{y} + \frac{y}{x} +2 = 9\\ \\ \Rightarrow \frac{x}{y} + \frac{y}{x} = 9-2\\ \\ \Rightarrow \boxed{\frac{x}{y} + \frac{y}{x} = 7}$