1. **State the problem:** Simplify the expression $$8 \log 4 \alpha^6 - 6 \log 7 x^3$$. 2. **Recall logarithm rules:** - Power rule: $$\log_b (a^n) = n \log_b a$$ - Product rule: $$\log_b (xy) = \log_b x + \log_b y$$ - Coefficients can be combined with the power rule. 3. **Apply the power rule to each logarithm:** $$8 \log 4 \alpha^6 = 8 (\log 4 + \log \alpha^6) = 8 \log 4 + 8 \log \alpha^6$$ $$6 \log 7 x^3 = 6 (\log 7 + \log x^3) = 6 \log 7 + 6 \log x^3$$ 4. **Apply the power rule inside the logs:** $$8 \log \alpha^6 = 8 \times 6 \log \alpha = 48 \log \alpha$$ $$6 \log x^3 = 6 \times 3 \log x = 18 \log x$$ 5. **Rewrite the expression:** $$8 \log 4 + 48 \log \alpha - 6 \log 7 - 18 \log x$$ 6. **Group terms:** $$(8 \log 4 - 6 \log 7) + (48 \log \alpha - 18 \log x)$$ 7. **Use the property $$a \log b - c \log d = \log b^a - \log d^c = \log \frac{b^a}{d^c}$$ to combine each group:** $$8 \log 4 - 6 \log 7 = \log 4^8 - \log 7^6 = \log \frac{4^8}{7^6}$$ $$48 \log \alpha - 18 \log x = \log \alpha^{48} - \log x^{18} = \log \frac{\alpha^{48}}{x^{18}}$$ 8. **Combine the two logarithms:** $$\log \frac{4^8}{7^6} + \log \frac{\alpha^{48}}{x^{18}} = \log \left( \frac{4^8}{7^6} \times \frac{\alpha^{48}}{x^{18}} \right) = \log \frac{4^8 \alpha^{48}}{7^6 x^{18}}$$ **Final simplified expression:** $$\boxed{\log \frac{4^8 \alpha^{48}}{7^6 x^{18}}}$$