Logarithm Evaluation 45377A

1. **State the problem:** We are given $\log_3 2 = 0.6309$ and $\log_3 5 = 1.465$ and need to evaluate: a) $\log_3 80$ b) $\log_3 1.28$ c) $\log_3 \frac{\sqrt{10}}{125}$ d) $\frac{\log_3 0.125}{\log_3 \sqrt{125}}$ 2. **Recall logarithm rules:** - $\log_b (xy) = \log_b x + \log_b y$ - $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ - $\log_b (x^r) = r \log_b x$ 3. **Calculate each part:** **a) Calculate $\log_3 80$** - Factorize $80 = 16 \times 5 = 2^4 \times 5$ - Use log rules: $$\log_3 80 = \log_3 (2^4 \times 5) = \log_3 2^4 + \log_3 5 = 4 \log_3 2 + \log_3 5$$ - Substitute values: $$= 4 \times 0.6309 + 1.465 = 2.5236 + 1.465 = 3.9886$$ - Round to 3 sf: $$\boxed{3.99}$$ **b) Calculate $\log_3 1.28$** - Express $1.28 = \frac{128}{100} = \frac{2^7}{10^2}$ - Since $10 = 2 \times 5$, $\log_3 10 = \log_3 2 + \log_3 5$ - Use log rules: $$\log_3 1.28 = \log_3 \left(\frac{2^7}{10^2}\right) = 7 \log_3 2 - 2 \log_3 10$$ - Substitute $\log_3 10 = \log_3 2 + \log_3 5 = 0.6309 + 1.465 = 2.0959$ - Calculate: $$= 7 \times 0.6309 - 2 \times 2.0959 = 4.4163 - 4.1918 = 0.2245$$ - Round to 3 sf: $$\boxed{0.225}$$ **c) Calculate $\log_3 \frac{\sqrt{10}}{125}$** - Express $\sqrt{10} = 10^{1/2}$ and $125 = 5^3$ - Use log rules: $$\log_3 \frac{\sqrt{10}}{125} = \log_3 10^{1/2} - \log_3 5^3 = \frac{1}{2} \log_3 10 - 3 \log_3 5$$ - Substitute values: $$= \frac{1}{2} \times 2.0959 - 3 \times 1.465 = 1.04795 - 4.395 = -3.34705$$ - Round to 3 sf: $$\boxed{-3.35}$$ **d) Calculate $\frac{\log_3 0.125}{\log_3 \sqrt{125}}$** - Express $0.125 = \frac{1}{8} = 2^{-3}$ - Express $\sqrt{125} = 125^{1/2} = (5^3)^{1/2} = 5^{3/2}$ - Calculate numerator: $$\log_3 0.125 = \log_3 2^{-3} = -3 \log_3 2 = -3 \times 0.6309 = -1.8927$$ - Calculate denominator: $$\log_3 \sqrt{125} = \log_3 5^{3/2} = \frac{3}{2} \log_3 5 = 1.5 \times 1.465 = 2.1975$$ - Divide: $$\frac{-1.8927}{2.1975} = -0.861$$ - Round to 3 sf: $$\boxed{-0.861}$$ 4. **Final answers:** - a) $3.99$ - b) $0.225$ - c) $-3.35$ - d) $-0.861$