1. **Problem:** Use laws of logarithms to combine the expression $$4\log 6 - 3\log 2 + 3\log 5 + 8\log 12 + \log 9$$. 2. **Formula and rules:** - Product rule: $$\log a + \log b = \log (ab)$$ - Quotient rule: $$\log a - \log b = \log \left(\frac{a}{b}\right)$$ - Power rule: $$k \log a = \log (a^k)$$ 3. **Apply power rule:** $$4\log 6 = \log (6^4) = \log 1296$$ $$-3\log 2 = \log (2^{-3}) = \log \frac{1}{8}$$ $$3\log 5 = \log (5^3) = \log 125$$ $$8\log 12 = \log (12^8)$$ (keep as is for now) $$\log 9$$ 4. **Combine all using product and quotient rules:** $$\log \left(\frac{1296 \times 125 \times 12^8 \times 9}{8}\right)$$ 5. **Simplify numerator and denominator:** - Numerator: $$1296 \times 125 \times 9 = 1296 \times 1125 = 1458000$$ - So expression is $$\log \left(\frac{1458000 \times 12^8}{8}\right)$$ 6. **Simplify fraction:** $$\frac{1458000}{8} = 182250$$ 7. **Final combined logarithm:** $$\log \left(182250 \times 12^8\right)$$ --- **Final answer:** $$\boxed{\log \left(182250 \times 12^8\right)}$$