1. **State the problem:** We need to find the exact value of $y$ given the equation $$7^{2y} = 3^{4}.$$ 2. **Recall the logarithm property:** To solve for $y$, we can take the logarithm of both sides. Using the property $$a^{b} = c \implies b = \log_{a} c,$$ we get $$2y = \log_{7} 3^{4}.$$ 3. **Simplify the logarithm:** Using the power rule of logarithms, $$\log_{a} b^{n} = n \log_{a} b,$$ we have $$2y = 4 \log_{7} 3.$$ 4. **Solve for $y$:** Divide both sides by 2 to isolate $y$: $$y = \frac{4}{2} \log_{7} 3 = 2 \log_{7} 3.$$ 5. **Conclusion:** The exact value of $y$ is $$y = 2 \log_{7} 3,$$ which corresponds to option (a).