Step-by-step Solution:

1. Given Equation: \[ (2x)^{\ln 2} = (3y)^{\ln 3} \] 2. Take the Natural Logarithm of Both Sides: \[ \ln\left((2x)^{\ln 2}\right) = \ln\left((3y)^{\ln 3}\right) \] \[ (\ln 2) \ln(2x) = (\ln 3) \ln(3y) \] \[ (\ln 2)(\ln 2 + \ln x) = (\ln 3)(\ln 3 + \ln y) \] 3. Simplify the Equation: \[ (\ln 2)\left(\frac{\ln 2}{\ln y} + \frac{\ln x}{\ln y}\right) = (\ln 3)\left(1 + \frac{\ln 3}{\ln y}\right) \] 4. Given Second Equation: \[ 3^{\ln x} = 2^{\ln y} \] 5. Take the Natural Logarithm of Both Sides: \[ \ln\left(3^{\ln x}\right) = \ln\left(2^{\ln y}\right) \] \[ (\ln x)(\ln 3) = (\ln y)(\ln 2) \] \[ \frac{\ln x}{\ln y} = \frac{\ln 2}{\ln 3} \] 6. Substitute and Solve: From the second equation, we have: \[ \frac{\ln x}{\ln y} = \frac{\ln 2}{\ln 3} \] Substitute into the first equation: \[ (\ln 2)\left(\frac{\ln 2}{\ln y} + \frac{\ln 2}{\ln 3}\right) = (\ln 3)\left(1 + \frac{\ln 3}{\ln y}\right) \] Simplify and solve for \( \ln y \): \[ \frac{(\ln 2)^2 - (\ln 3)^2}{\ln y} = \frac{(\ln 3)^2 (\ln 2)^2}{\ln 3} \] \[ \ln y = -\ln 3 \] \[ y = \frac{1}{3} \] 7. Find \( \ln x \): From the second equation: \[ \ln x = -\ln 2 \] \[ x = \frac{1}{2} \] Therefore, the solutions are: \[ \boxed{x = \frac{1}{2}} \] \[ \boxed{y = \frac{1}{3}} \]