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Answer 1

$if \: log( \frac{x - y}{2} ) = \frac{1}{2} ( log(x) + log(y) ) \\ \: prove \: that \: {x}^{2} + {y}^{2} = 6xy \\ solution : \\ log( \frac{x - y}{2} ) = \frac{1}{2} ( log(x) + log(y) ) \\ 2 log( \frac{x - y}{2} ) = log(xy) \\ log( {( \frac{x - y}{2} )}^{2} ) = log(xy) \\ {( \frac{x - y}{2} )}^{2} = xy \\ \frac{ {x}^{2} + {y}^{2} - 2xy}{4} = xy \\ {x}^{2} + {y}^{2} - 2xy = 4xy \\ {x}^{2} + {y}^{2} = 4xy + 2xy \\ {x}^{2} + {y}^{2} = 6xy \\ hence \: proved$


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