Let ( x 0 , y 0 ) be the solution of the following equations ( 2 x ) ln 2 = ( 3 y ) ln 3 ; 3 ln x = 2 ln y . Then x 0 is [2011]

Question

Let $(x_{0}, y_{0})$ be the solution of the following equations ${(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}; 3^{\ln x} = 2^{\ln y} .$ Then $x_{0}$ is [2011]

Smart Achievers · Accepted Answer

(3)

$Given : {(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}$

$\Rightarrow \ln 2 \cdot \ln (2 x) = \ln 3 \cdot \ln (3 y)$

$\Rightarrow \ln 2 \cdot \ln (2 x) = \ln 3 \cdot (\ln 3 + \ln y) \dots (i)$

$Also given : 3^{\ln x} = 2^{\ln y}$

$\Rightarrow \ln x \cdot \ln 3 = \ln y \cdot \ln 2 \Rightarrow \ln y = \frac{\ln x \cdot \ln 3}{\ln 2} \dots (i i)$

$From equation (i) and (ii), we get$

$\ln 2 \cdot \ln (2 x) = \ln 3 [\ln 3 + \frac{\ln x \cdot \ln 3}{\ln 2}]$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) = {(\ln 3)}^{2} \ln 2 + {(\ln 3)}^{2} \ln x$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) = {(\ln 3)}^{2} (\ln 2 + \ln x)$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) - {(\ln 3)}^{2} \ln (2 x) = 0$

$\Rightarrow [{(\ln 2)}^{2} - {(\ln 3)}^{2}] \ln (2 x) = 0$ $\Rightarrow \ln (2 x) = 0$

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

Solution

Q.
Let $(x_{0}, y_{0})$ be the solution of the following equations ${(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}; 3^{\ln x} = 2^{\ln y} .$ Then $x_{0}$ is [2011]

1 $\frac{1}{6}$

2 $\frac{1}{3}$

3 $\frac{1}{2}$

4 $6$

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Solution

Q. Let (x0,y0) be the solution of the following equations (2x)ln2=(3y)ln3; 3lnx=2lny. Then x0 is [2011]

1 16

2 13

3 12