1. **State the problem:** Solve the simultaneous equations: $$\ln\left(\frac{y}{x}\right) = 2$$ $$\ln x^{2} + \ln y^{3} = 7$$ 2. **Recall logarithm properties:** - $\ln\left(\frac{a}{b}\right) = \ln a - \ln b$ - $\ln a + \ln b = \ln(ab)$ - $\ln a^{m} = m \ln a$ 3. **Rewrite the first equation:** $$\ln y - \ln x = 2$$ 4. **Rewrite the second equation using properties:** $$\ln x^{2} + \ln y^{3} = \ln(x^{2} y^{3}) = 7$$ 5. **From the first equation, express $\ln y$ in terms of $\ln x$:** $$\ln y = 2 + \ln x$$ 6. **Substitute $\ln y$ into the second equation:** $$\ln(x^{2} y^{3}) = 7$$ $$\ln(x^{2}) + \ln(y^{3}) = 7$$ $$2 \ln x + 3 \ln y = 7$$ Substitute $\ln y = 2 + \ln x$: $$2 \ln x + 3(2 + \ln x) = 7$$ 7. **Simplify:** $$2 \ln x + 6 + 3 \ln x = 7$$ $$5 \ln x + 6 = 7$$ 8. **Isolate $\ln x$:** $$5 \ln x = 7 - 6$$ $$5 \ln x = 1$$ $$\ln x = \frac{1}{5}$$ 9. **Solve for $x$:** $$x = e^{\frac{1}{5}}$$ 10. **Find $y$ using $\ln y = 2 + \ln x$:** $$\ln y = 2 + \frac{1}{5} = \frac{10}{5} + \frac{1}{5} = \frac{11}{5}$$ $$y = e^{\frac{11}{5}}$$ **Final answer:** $$x = e^{\frac{1}{5}}, \quad y = e^{\frac{11}{5}}$$