1. **Stating the problem:** We are given an expression involving logarithms with base 7: $$\log_7 174 = 7 \log_7 174 + 7 \log_7 174$$ and need to analyze or simplify it. 2. **Recall logarithm properties:** - The logarithm of a product: $$\log_b (xy) = \log_b x + \log_b y$$ - The logarithm of a power: $$\log_b (x^k) = k \log_b x$$ 3. **Analyze the given equation:** $$\log_7 174 = 7 \log_7 174 + 7 \log_7 174$$ 4. **Combine the right side:** $$7 \log_7 174 + 7 \log_7 174 = 14 \log_7 174$$ 5. **Rewrite the equation:** $$\log_7 174 = 14 \log_7 174$$ 6. **Subtract $$\log_7 174$$ from both sides:** $$\log_7 174 - 14 \log_7 174 = 0$$ 7. **Factor out $$\log_7 174$$:** $$\cancel{\log_7 174} (1 - 14) = 0$$ 8. **Simplify:** $$\log_7 174 \times (-13) = 0$$ 9. **Since $$-13 \neq 0$$, for the product to be zero:** $$\log_7 174 = 0$$ 10. **Check if $$\log_7 174 = 0$$ is true:** By definition, $$\log_7 174 = 0$$ means $$7^0 = 174$$ which is $$1 = 174$$, which is false. **Conclusion:** The given equation is not true for any real value; it is a contradiction. **Final answer:** The equation $$\log_7 174 = 7 \log_7 174 + 7 \log_7 174$$ is false and has no solution.