Question

if log x+y/2 = 1/2(logx+logy) prove that x=y

Answer 1

$log( \frac{x + y}{2} ) = \frac{1}{2} ( log(x) + log(y) ) \\ or. \: 2 log( \frac{x + y}{2} ) = log(x) + log(y) \\ or. \: { log( \frac{x + y}{2} ) }^{2} = log(xy) \\ or. \: \frac{ {(x + y)}^{2} }{4} = xy \\ or. \: {x}^{2} + 2xy + {y}^{2} = 4xy \\ or. \: {x}^{2} - 2xy + {y}^{2} = 0 \\ or. \: {(x - y)}^{2} = 0 \\ or. \: x - y = 0 \\ or. \: x = y.....{proved}$
Hence ur answer is proved.
Hope I helped you.

Answer 2

Answer:

Step-by-step explanation:

The given equation is:

$log(\frac{x+y}{2})=\frac{1}{2}(logx+logy)$

On simplifying, we get

$2log(\frac{x+y}{2})=logx+logy$

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

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

$x^2+2xy+y^2=4xy$

$x^2-2xy+y^2=0$

$(x-y)^{2}=0$

$x-y=0$

$x=y$

Hence proved.