Question:

Step 1: Convert all logarithms to base 3. Let $\log_3 x = a$ and $\log_3 y = b$. $$\log_9 x = \frac{a}{2}, \quad \log_{27} x = \frac{a}{3}, \quad \log_9 y = \frac{b}{2}, \quad \log_{27} y = \frac{b}{3}$$ Step 2: Substitute into the two equations: $$\frac{a}{2} + \frac{b}{3} = \frac{7}{2} \quad \Rightarrow \quad 3a + 2b = 21 \quad \cdots (1)$$ $$\frac{a}{3} + \frac{b}{2} = \frac{2}{3} \quad \Rightarrow \quad 2a + 3b = 4 \quad \cdots (2)$$ Step 3: Solve the system of equations. From (1): $3a + 2b = 21$ From (2): $2a + 3b = 4$ Multiply (1) by 3 and (2) by 2: $$9a + 6b = 63$$ $$4a + 6b = 8$$ Subtracting: $5a = 55 \Rightarrow a = 11$ Substitute $a = 11$ into (2): $22 + 3b = 4 \Rightarrow b = -6$ Step 4: Find $xy$ and $x/y$. $$\log_3(xy) = a + b = 11 + (-6) = 5 \Rightarrow xy = 3^5 = 243$$ $$\log_3\left(\frac{x}{y}\right) = a - b = 11 - (-6) = 17 \Rightarrow \frac{x}{y} = 3^{17}$$ Step 5: Check options: $xy = 243$ ✓ (option 1), $\dfrac{x}{y} = 3^{17}$ ✓ (option 4).