If $x^{2} + y^{2} = 25 x y,$ then prove that $2 l o g (x + y) = 3 l o g 3 + l o g x + l o g y$

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Solution

Given

$x^{2} + y^{2} = 25 x y$ ....(1)

$L . H . S = 2 l o g (x + y)$

$= l o g (x + y)^{2}$

$= l o g (x^{2} + y^{2} + 2 x y)$

$= l o g (25 x y + 2 x y)$ [ from (1)]

$= l o g (27 x y)$

$= l o g (3^{3} x y)$

$= l o g 3^{3} + l o g x + l o g y$

$= 3 l o g 3 + l o g x + l o g y$

$= R . H . S$

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