If(x,y)  is the solution of the following equations (2x)^log2= (3y)^log3 and 3^logx = 2^logy then x is equal to?

Jun 3, 2018

$x = \frac{1}{2} , y = \frac{1}{3}$

Explanation:

Taking the logarthm on both sides (the first equation) we get
$\log \left(2\right) \left(\log \left(2\right) + \log \left(x\right)\right) = \log \left(3\right) \left(\log \left(3\right) + \log \left(y\right)\right)$

Doing the same with the second equation:

$\log \left(y\right) = \log \left(x\right) \cdot \log \frac{3}{\log} \left(2\right)$
Substituting

$a = \log \left(x\right)$

we get

$\log \left(x\right) \frac{{\log}^{2} \left(2\right) - {\log}^{2} \left(3\right)}{\log} \left(2\right) = - \left({\log}^{2} \left(2\right) - {\log}^{2} \left(3\right)\right)$

so $a = - \log \left(2\right)$

$\log \left(x\right) = \log \left({2}^{- 1}\right)$

$x = \frac{1}{2}$

In the first equation we get

$1 = 3 y$

$y = \frac{1}{3}$