Let $\left(x_0, y_0\right)$ 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 equal to

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Aicharya · Accepted Answer

Taking ln on both sides, $\ln 2 \cdot \ln (2 x)=\ln 3(\ln 3 y)$ $\begin{array}{ll} \Rightarrow & \ln 2\{\ln 2+\ln x\} \ & =\ln 3\{\ln 3+\ln y\} \ 	ext { and } & \ln x \cdot \ln 3=\ln y \ln 2 \ & \ln y=\frac{\ln x \cdot \ln 3}{\ln 2} \end{array}$ From Eqs. (i) and (ii), we get $\begin{aligned} & \ln 2\{\ln 2+\ln x\} \ &=\ln 3 \cdot\left\{\ln 3+\frac{\ln x \cdot \ln 3}{\ln 2}ight\} \ & \Rightarrow(\ln 2)^2+\ln 2 \cdot \ln x \ &=(\ln 3)^2+\frac{(\ln 3)^2}{(\ln 2)} \cdot \ln x \ & \Rightarrow \ln x\left\{\frac{(\ln 3)^2}{\ln 2}-\ln 2ight\} \ &=(\ln 2)^2-(\ln 3)^2 \end{aligned}$ $\begin{aligned} & \Rightarrow \ln x=-\ln 2=\ln 2^{-1} \ & 	herefore x=\frac{1}{2} \end{aligned}$

Let $\left(x_0, y_0\right)$ be the solution of the following equations $(2 x)^{\ln 2}=(3 y)^{\ln 3}$,…

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