1. **Stating the problem:** Simplify the expression $\log_{7} 77x - 20\log_{7} x - 5\log_{7} (x+1)$. 2. **Recall logarithm properties:** - $\log_b (MN) = \log_b M + \log_b N$ - $k \log_b M = \log_b M^k$ - $\log_b \frac{M}{N} = \log_b M - \log_b N$ 3. **Apply the product rule to $\log_7 77x$:** $$\log_7 77x = \log_7 77 + \log_7 x$$ 4. **Rewrite the expression:** $$\log_7 77 + \log_7 x - 20 \log_7 x - 5 \log_7 (x+1)$$ 5. **Combine like terms for $\log_7 x$:** $$\log_7 77 + (1 - 20) \log_7 x - 5 \log_7 (x+1) = \log_7 77 - 19 \log_7 x - 5 \log_7 (x+1)$$ 6. **Rewrite coefficients as exponents:** $$\log_7 77 - \log_7 x^{19} - \log_7 (x+1)^5$$ 7. **Use subtraction rule for logarithms:** $$\log_7 \frac{77}{x^{19}} - \log_7 (x+1)^5 = \log_7 \frac{77}{x^{19} (x+1)^5}$$ **Final answer:** $$\boxed{\log_7 \frac{77}{x^{19} (x+1)^5}}$$