Question

If log(x+y/3) = 1/2(logx + logy), then find the value of 1/x-1/y.

Answer 1

$log( \frac{x+y}{2} )= \frac{1}{2}(logx+logy)\\ \\ \ log( \frac{x+y}{2} )= log \sqrt{xy} \\ \\ \frac{x+y}{2}= \sqrt{xy} \\ \\ x+y=2 \sqrt{xy} \\ \\(x+y)^2=(2 \sqrt{xy})^2 \\ \\ x^{2} +y^2+2xy=4xy\\ \\x^{2} +y^2-2xy=0\\ \\(x-y)^2=0\\ \\x-y = 0\\ \\dividing\ by\ xy\\ \\ \frac{x}{xy}- \frac{y}{xy}= \frac{0}{xy}\\ \\ \frac{1}{y}- \frac{1}{x}=0 \\ \\\frac{1}{x}- \frac{1}{y}=0$