Question

Logx+y/5=1/2(logx+logy) then proved x/y+y/x=23

Answer 1

Answer:

Logarithmic identities used:-)

$log(m) + log(n) = log(m.n) \\ \\ n \times log(m) = log( {m}^{n} )$

Solution:-)

Given that,

$\: \: \: \: \: \: \: \: \: \: \: log( \dfrac{x + y}{5} ) = \dfrac{1}{2} \times ( log \: x + log \: y) \\ \\ \implies log( \frac{x + y}{5} ) = \frac{1}{2} \times log(x.y) \\ \\ \implies log( \frac{x + y}{5} ) = log {(x.y)}^{ \frac{1}{2} } \\ \\ \implies \frac{x + y}{5} = ( {x.y)}^{ \frac{1}{2} } \\ \\ \implies \frac{x + y}{5} = \sqrt{x.y} \\ \\ \implies \frac{x + y}{ \sqrt{x.y} } = 5 \\ \\ \implies \frac{x }{ \sqrt{x.y} } + \frac{ y}{ \sqrt{x.y} } = 5\\ \\ \implies \sqrt{ \frac{x}{y} } + \sqrt{ \frac{y}{x} } = 5 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: s.b.s \\ \\ \implies \frac{x}{y} + \frac{y}{x} + 2 \sqrt{ \frac{x}{y} \times { \frac{y}{x} } } = 25 \\ \\ \implies \frac{x}{y} + \frac{y}{x} + 2 = 25 \\ \\ \implies \frac{x}{y} + \frac{y}{x + } = 23$

Hence the required result is proved .