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

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

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Answer 1 · 2018-06-03T07:07:43

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

Explanation:

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

Doing the same with the second equation:

$log (y) = log (x) \cdot \frac{log (3)}{log (2)}$ $log(y)=log(x)*log(3)/log(2)$
Substituting

$a = log (x)$ $a=log(x)$

we get

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

so $a = - log (2)$ $a=-log(2)$

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

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

In the first equation we get

$1 = 3 y$ $1=3y$

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

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

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Explanation:

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If(x,y) (x,y)(x,y) is the solution of the following equations (2x)^log2= (3y)^log3(2x)log2=(3y)log3(2x)^log2= (3y)^log3 and 3^logx = 2^logy3logx=2logy3^logx = 2^logy then x is equal to?

1 Answer

Explanation:

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