Question

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Answer 1

Question :-

If log [ (x + y)/3 ] = 1/2 [log x + log y ], then find the value of x/y + y/x.

Solution :-

$log \dfrac{x + y}{3} = \dfrac{1}{2} \bigg(log \ x + log \ y \bigg)$

$\implies 2log \dfrac{x + y}{3} = log \ x + log \ y$

$\implies log \bigg( \dfrac{x + y}{3} \bigg)^{2} = log \ x + log \ y$

[ Because n.log x = log x^n ]

$\implies log \bigg( \dfrac{x + y}{3} \bigg)^{2} = log \ xy$

[ Because log x + log y = log xy ]

Comparing on both sides

$\implies \bigg( \dfrac{x + y}{3} \bigg)^{2} = xy$

$\implies \dfrac{(x + y)^{2} }{3^{2} } = xy$

$\implies \dfrac{ {x}^{2} + {y}^{2} + 2xy }{9 } = xy$

[ Because (x + y)² = x² + y² + 2xy ]

⇒ x² + y² + 2xy = 9xy

⇒ x² + y² = 9xy - 2xy

⇒ x² + y² = 7xy

$\implies \dfrac{ {x}^{2} + {y}^{2}}{xy } = 7$

It can be written as :

$\implies \dfrac{ {x}^{2} }{xy } + \dfrac{ {y}^{2} }{xy} = 7$

$\implies \boxed{\dfrac{x }{y } + \dfrac{ y}{x} = 7}$

Hence, the value of x/y + y/x is 7.

Answer 2

${\large\bf{\mid{\overline{\underline{Given:-}}}\mid}}$

• log(x+y)/3 = 1/2(logx + logy)

$\Large\underline\mathfrak{Question}$

• x/y + y/x ?

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

__________________

$\red{\bold{we\:know\:that:--}}$

● $\orange{\bold{Product\:Rule\:Law:}}$

loga (MN) = loga M + loga N

● $\green{\bold{Power\:Rule\:Law:}}$

IogaM^(n) = n Ioga M

● (x+y)² = x² + y² + 2xy

_________________________

$\boxed{\fcolorbox{white}{pink}{ANSWER:-}}$

$log( \frac{x + y}{3} ) = \frac{1}{2} ( log(x) + log(y) ) \\ \\ \red\leadsto \: 2log( \frac{x + y}{3} ) = ( log(x) + log(y) ) \\ \\ \:\red\leadsto \: (log \frac{x + y}{3} )^{2} = log(x \times y) \\ \\ \red{\textbf{comparing \: now}} \\ \\ \red\leadsto \: \frac{(x + y)^{2} }{9} = {xy}^{2} \\ \\ \red\leadsto \: {x}^{2} + {y}^{2} + 2xy = 9 {xy} \\ \\ \red\leadsto \: {x}^{2} + {y}^{2} = 9xy - 2xy \\ \\ \red\leadsto {x}^{2} + {y}^{2} = 7xy \\ \\ \green{\text{Dividing by xy both sides, we get,}} \\ \\ \red\leadsto \: \frac{ {x}^{ \cancel2} }{\cancel{x}y} + \frac{ {y}^{ \cancel2} }{ x\cancel{y}} = \frac{7 \cancel{xy}}{ \cancel{xy}} \\ \\ \red\leadsto \: \pink{\large\boxed{\bold{\frac{x}{y} + \frac{y}{x} = 7}}}$

$\large\underline\textbf{Hope it Helps You.}$