Question

Let $\left(x_{0}, y_{0}\right)$ be solution of the following equations
$(2 x )^{\ln 2} =(3 y )^{\ln 3}$
$3^{\ln x} \&=2^{\ln y}$ Then $x_{0}$ is

• A. $\frac{1}{6}$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. 6

Answer 1

Answer: (C)

$(2 x )^{\ln 2}=(3 y )^{\ln 3}$ ...(i)
$3^{\ln x}=2^{lny}$ ...(ii)
$\Rightarrow (\log x)(\log 3)=(\log y) \log 2$
$\Rightarrow \log y=\frac{(\log x)(\log 3)}{\log 2}$ ...(iii)
In (i) taking $\log$ both sides
$(\log 2)\{\log 2+\log \}=\log 3\{\log 3+\log y\}$
$(\log 2)^{2}+(\log 2)(\log x)=(\log 3)^{2}+\frac{(\log 3)^{2}(\log x)}{\log 2}$ from (iii) $\Rightarrow (\log 2)^{2}-(\log 3)^{2}=\frac{(\log 3)^{2}-(\log 2)^{2}}{\log 2}(\log x)$
$\Rightarrow -\log 2=\log x$
$\Rightarrow x=\frac{1}{2}$
$\Rightarrow x_{0}=\frac{1}{2}$