(x_o, y_o) be the solution of the following equations $(2\mathrm{x}{)}^{\ln 2}=(3\mathrm{y}{)}^{\ln 3},3^{\ln \mathrm{x}}=2^{\ln \mathrm{y}}$ then x₀ is

$\begin{array}{l}(2\mathrm{x}{)}^{\ln 2}=(3\mathrm{y}{)}^{\ln 3},3^{\ln \mathrm{x}}=2^{\ln \mathrm{y}}\\ \Rightarrow \ln 2(\ln 2+\ln \mathrm{x})=\ln 3(\ln 3+\ln \mathrm{y}),\ln \mathrm{x}\cdot \ln 3=\ln \mathrm{y}\cdot \ln 2\\ \Rightarrow (\ln 2{)}^2++\ln 2\cdot \ln \mathrm{x}=(\ln 3{)}^2+\ln 3\left(\frac{\ln \mathrm{xln}3}{\ln 2}\right)\\ \Rightarrow \ln \mathrm{x}\left(\ln 2-\frac{(\ln 3{)}^2}{\ln 2}\right)=(\ln 3{)}^2-(\ln 2{)}^2\\ \Rightarrow \frac{\ln \mathrm{x}}{\ln 2}=-1\Rightarrow \mathrm{x}=\frac{1}{2}\end{array}$