Question

If x2+y2=14xy then prove that log(x+y)/4=1/4x+1/4y

Answer 1

$Given \: x^{2} + y^{2} = 14xy$

/* Add bothsides of the equation by 2xy, we get */

$\implies x^{2} + y^{2} + 2xy = 14xy + 2xy$

$\implies ( x + y )^{2} = 16xy$

$\implies \big(\frac{ x + y }{4}\big)^{2} = xy$

$\implies log \big(\frac{x + y }{4}\big)^{2} = log (xy)$

$\implies 2log \big(\frac{ x + y }{4}\big) = log x + log y$

$\implies log \big(\frac{x + y }{4}\big) =\frac{1}{2}\big( log x + log y\big)$

$\implies log \big(\frac{( x + y )}{4}\big) =\frac{1}{2} log x + \frac{1}{2} log y$

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