If $x^{2} + y^{2} - 25 x y, t h e n p r o v e 2 l o g (x + y) = 3 l o g 3 + l o g x + l o g y$

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Solution

$x^{2} + y^{2} = 25 x y \Rightarrow x^{2} + y^{2} + 2 x y = 25 x y + 2 x y (A d d i n g 2 x y o n b o t h s i d e s) \Rightarrow (x + y)^{2} = 27 x y \Rightarrow l o g (x + y)^{2} = l o g 27 x y (t a k i n g l o g o n b o t h s i d e s) \Rightarrow 2 l o g (x + y) = l o g 27 + l o g x + l o g y (l o g a b c = l o g a + l o g b + l o g c) \Rightarrow 2 l o g (x + y) = l o g 3^{3} + l o g x + l o g y \Rightarrow 2 l o g (x + y) = 3 l o g 3 + l o g x + l o g y$

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