1. The problem is to verify the logarithmic identity: $$\ln(xy) = \ln x - \ln y$$. 2. Recall the logarithm product rule: $$\ln(ab) = \ln a + \ln b$$ for positive $a$ and $b$. 3. Using this rule, $$\ln(xy) = \ln x + \ln y$$, which is different from the given expression. 4. The given expression $$\ln(xy) = \ln x - \ln y$$ is incorrect unless $y$ is replaced by $\frac{1}{y}$. 5. If we consider $$\ln\left(x \cdot \frac{1}{y}\right) = \ln x + \ln \frac{1}{y} = \ln x - \ln y$$, then the identity holds. 6. Therefore, the correct identity is $$\ln\left(\frac{x}{y}\right) = \ln x - \ln y$$. 7. In conclusion, $$\ln(xy) = \ln x + \ln y$$ and $$\ln\left(\frac{x}{y}\right) = \ln x - \ln y$$. This clarifies the correct logarithmic properties.