1. **State the problem:** Simplify the expressions: $$A = \ln\left(\frac{1}{3}\right) - \ln(27) + \ln\left(8 \frac{1}{2}\right)$$ $$B = \ln\left(\frac{35}{12}\right) + \ln\left(\frac{6}{7}\right) + \ln\left(\frac{4}{5}\right)$$ $$C = \ln(\sqrt{3} - \sqrt{2}) + \ln(\sqrt{3} + \sqrt{2})$$ 2. **Recall logarithm rules:** - $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ - $\ln(a) + \ln(b) = \ln(ab)$ - $\ln(\sqrt{x}) = \frac{1}{2} \ln(x)$ 3. **Simplify A:** - Convert mixed number: $8 \frac{1}{2} = 8.5 = \frac{17}{2}$ - Use subtraction rule: $\ln\left(\frac{1}{3}\right) - \ln(27) = \ln\left(\frac{1/3}{27}\right) = \ln\left(\frac{1}{81}\right)$ - So, $A = \ln\left(\frac{1}{81}\right) + \ln\left(\frac{17}{2}\right) = \ln\left(\frac{1}{81} \times \frac{17}{2}\right) = \ln\left(\frac{17}{162}\right)$ 4. **Simplify B:** - Use addition rule: $B = \ln\left(\frac{35}{12} \times \frac{6}{7} \times \frac{4}{5}\right)$ - Multiply numerators: $35 \times 6 \times 4 = 840$ - Multiply denominators: $12 \times 7 \times 5 = 420$ - So, $B = \ln\left(\frac{840}{420}\right) = \ln(2)$ 5. **Simplify C:** - Use addition rule: $C = \ln\left((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})\right)$ - Multiply conjugates: $(a - b)(a + b) = a^2 - b^2$ - So, $C = \ln(3 - 2) = \ln(1) = 0$ **Final answers:** $$A = \ln\left(\frac{17}{162}\right), \quad B = \ln(2), \quad C = 0$$