1. **State the problem:** Simplify or evaluate the expression $$7) \log_c \frac{a+1}{b^8}$$. 2. **Recall the logarithm rule:** The logarithm of a quotient is the difference of the logarithms: $$\log_c \frac{x}{y} = \log_c x - \log_c y$$ 3. **Apply the rule:** $$7) \log_c \frac{a+1}{b^8} = 7) \left( \log_c (a+1) - \log_c (b^8) \right)$$ 4. **Use the power rule of logarithms:** $$\log_c (b^8) = 8 \log_c b$$ 5. **Substitute back:** $$7) \left( \log_c (a+1) - 8 \log_c b \right) = 7 \log_c (a+1) - 56 \log_c b$$ 6. **Final simplified expression:** $$7 \log_c (a+1) - 56 \log_c b$$ This is the simplified form of the given logarithmic expression.