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:** $$4\log 6 = \log 6^4$$ $$-3\log 2 = \log 2^{-3}$$ $$3\log 5 = \log 5^3$$ $$8\log 12 = \log 12^8$$ $$\log 9 = \log 9$$ 4. **Rewrite expression:** $$\log 6^4 + \log 5^3 + \log 12^8 + \log 9 + \log 2^{-3}$$ 5. **Combine using product and quotient rules:** $$\log \left(\frac{6^4 \cdot 5^3 \cdot 12^8 \cdot 9}{2^3}\right)$$ 6. **Calculate powers:** $$6^4 = 1296$$ $$5^3 = 125$$ $$12^8 = 429981696$$ $$9 = 9$$ $$2^3 = 8$$ 7. **Substitute values:** $$\log \left(\frac{1296 \times 125 \times 429981696 \times 9}{8}\right)$$ 8. **Multiply numerator:** $$1296 \times 125 = 162000$$ $$162000 \times 429981696 = 69656014752000$$ $$69656014752000 \times 9 = 626904132768000$$ 9. **Divide by denominator:** $$\frac{626904132768000}{8} = 78363016596000$$ 10. **Final combined logarithm:** $$\boxed{\log 78363016596000}$$