Question

if Xsquare +Ysquare=6xy prove that 2log(x+y) =log x+log y+3log2

Answer 1

Answer:

$given \\ {x}^{2} + {y}^{2} = 6xy \\ add \: 2xy \: on \: both \: sides \\ {x}^{2} + {y}^{2} + 2xy = 6xy + 2xy \\ ( {x + y})^{2} = 8xy \\ add \: log \: on \: both \: sides \\ log( {x + y})^{2} = log8xy \\ from \: an \: identity \\ log {n}^{x} = x logn \\ 2 log(x + y) = log8xy \\ from \: this \: identity \\ logab = loga + logb \\ 2 log(x + y) = log {2}^{3} + logx + logy \\ 2 log(x + y) = 3 log2 + logx + logy$

hence proved

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