QJEEAdv11Ma5a5b: JEE Advanced 2011 Mathematics - Class 11 JEEAdv11Ma5a5b Solution: Let be the solution of the following equations ( 2 x ) ℓ n 2 =…

We have,

${(2x)^{\ln 2}} = {(3y)^{\ln 3}}$ ...... (1)

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

$\Rightarrow (\log x)(\log 3) = (\log y)\log 2$

$\Rightarrow \log y = {{(\log x)(\log 3)} \over {\log 2}}$ ........ (3)

Taking log both sides of Eq. (1), we get

$(\log 2)\{ \log 2 + \log x\} = \log 3\{ \log 3 + \log y\}$

${(\log 2)^2} + (\log 2)(\log x) = {(\log 3)^2} + {{{{(\log 3)}^2}(\log x)} \over {\log 2}}$ from Eq. (3)

$\Rightarrow {(\log 2)^2} - {(\log 3)^2} = {{{{(\log 3)}^2} - {{(\log 2)}^2}} \over {\log 2}}(\log x)$

$\Rightarrow - \log 2 = \log x$

$\Rightarrow x = {1 \over 2} \Rightarrow {x_0} = {1 \over 2}$ .