1. **State the problem:** We need to find the value of $\frac{y}{x}$ given the equation $9^x = 81^y$. 2. **Rewrite the bases as powers of the same base:** Note that $9 = 3^2$ and $81 = 3^4$. So, the equation becomes: $$ (3^2)^x = (3^4)^y $$ 3. **Use the power of a power rule:** $$ 3^{2x} = 3^{4y} $$ 4. **Since the bases are equal, set the exponents equal:** $$ 2x = 4y $$ 5. **Solve for $\frac{y}{x}$:** $$ \frac{y}{x} = \frac{2x}{4x} = \frac{2}{4} = \frac{1}{2} $$ **Final answer:** $$ \boxed{\frac{y}{x} = \frac{1}{2}} $$