1. **Problem:** Simplify the expression $2 \ln(a^3 b^4) - 3 \ln a - 3 \ln(ab^2)$ as simply as possible. 2. **Recall logarithm rules:** - $\ln(xy) = \ln x + \ln y$ - $\ln(x^k) = k \ln x$ - Coefficients can be brought inside as exponents: $c \ln x = \ln(x^c)$ 3. **Apply the rules:** $$2 \ln(a^3 b^4) = 2 (\ln a^3 + \ln b^4) = 2 (3 \ln a + 4 \ln b) = 6 \ln a + 8 \ln b$$ 4. Substitute back: $$6 \ln a + 8 \ln b - 3 \ln a - 3 (\ln a + 2 \ln b)$$ 5. Distribute the last term: $$6 \ln a + 8 \ln b - 3 \ln a - 3 \ln a - 6 \ln b$$ 6. Combine like terms: $$ (6 \ln a - 3 \ln a - 3 \ln a) + (8 \ln b - 6 \ln b) = 0 \ln a + 2 \ln b = 2 \ln b$$ 7. Final simplified form: $$2 \ln b$$ **Answer:** $2 \ln b$