Question

Solution: 10.If 2log((x+y)/(4))=log x+log y then find the value of (x)/(y)+(y)/(x)​

Answer 1

SOLUTION :

GIVEN

$\displaystyle \sf{ 2\log \bigg( \frac{x + y}{4} \bigg) = \log x + \log y}$

TO DETERMINE

$\displaystyle \sf{ \: } The \: value \: of \: \bigg( \frac{x}{y} + \frac{y}{x} \bigg)$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ \: } 1. \: \: \log {x}^{m} = m \log x$

$\displaystyle \sf{ \: } 2. \: \: \log {(xy)} = \log x + \log y$

CALCULATION

$\displaystyle \sf{ 2\log \bigg( \frac{x + y}{4} \bigg) = \log x + \log y}$

$\implies \displaystyle \sf{ \log {\bigg( \frac{x + y}{4} \bigg)}^{2} = \log (x y)}$

$\implies \displaystyle \sf{ {\bigg( \frac{x + y}{4} \bigg)}^{2} = x y}$

$\implies \displaystyle \sf{ ( {x + y)}^{2} = 16 x y}$

$\implies \displaystyle \sf{ {x}^{2} + 2xy + {y}^{2} = 16 x y}$

$\implies \displaystyle \sf{ {x}^{2} + {y}^{2} = 14 x y}$

Dividing both sides by xy we get

$\implies \displaystyle \sf{ \frac{ {x}^{2} }{xy} + \frac{ {y}^{2} }{xy} = 14}$

$\implies \displaystyle \sf{ \: } \frac{x}{y} + \frac{y}{x} = 14$

RESULT

$\boxed {\displaystyle \sf{ \: } \: \: \frac{x}{y} + \frac{y}{x} = 14 \: \: }$

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LEARN MORE FROM BRAINLY

IF 1/1!9!+1/3!7!+1/5!5!+1/7!3!+1/9!=2^n /10!

then n=?

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