1. **State the problem:** Simplify the expression $\log_7 (x^7 y^2)$. 2. **Recall the logarithm product rule:** $\log_b (MN) = \log_b M + \log_b N$. This means we can split the log of a product into the sum of logs. 3. **Apply the product rule:** $$\log_7 (x^7 y^2) = \log_7 (x^7) + \log_7 (y^2)$$ 4. **Recall the power rule for logarithms:** $\log_b (M^k) = k \log_b M$. This allows us to bring exponents in front as multipliers. 5. **Apply the power rule:** $$\log_7 (x^7) + \log_7 (y^2) = 7 \log_7 x + 2 \log_7 y$$ 6. **Final simplified expression:** $$7 \log_7 x + 2 \log_7 y$$ This is the simplest form, expressing the logarithm of the product as a sum of logarithms without exponents on variables.