1. **Problem a:** Write the scientific form of 0.006237 and find the characteristic of its logarithm. - First, write 0.006237 in scientific notation. Move the decimal 3 places right: $$0.006237 = 6.237 \times 10^{-3}$$ - The characteristic of the logarithm of a number less than 1 is the negative integer part corresponding to the exponent, so the characteristic is $$-3$$. 2. **Problem b:** Determine the value of the expression $$\log_7 \sqrt{7} - \log_3 3\sqrt{3} + \log_5 5\sqrt{5} + \log_2 \sqrt{2}$$. - Recall that $$\log_a (a^k) = k$$ and $$\sqrt{a} = a^{1/2}$$. - Evaluate each term: 1) $$\log_7 \sqrt{7} = \log_7 7^{1/2} = \frac{1}{2}$$ 2) $$\log_3 3\sqrt{3} = \log_3 (3 \times 3^{1/2}) = \log_3 3^{3/2} = \frac{3}{2}$$ 3) $$\log_5 5\sqrt{5} = \log_5 (5 \times 5^{1/2}) = \log_5 5^{3/2} = \frac{3}{2}$$ 4) $$\log_2 \sqrt{2} = \log_2 2^{1/2} = \frac{1}{2}$$ - Sum them up: $$\frac{1}{2} - \frac{3}{2} + \frac{3}{2} + \frac{1}{2} = \left(\frac{1}{2} + \frac{1}{2}\right) + \left(-\frac{3}{2} + \frac{3}{2}\right) = 1 + 0 = 1$$ 3. **Problem c:** Given $$a^x = b$$, $$b^y = c$$, and $$c^z = a$$, prove that $$xyz$$ equals the value of the expression from (b). - Taking log on both sides of each: 1) $$x \log a = \log b$$ 2) $$y \log b = \log c$$ 3) $$z \log c = \log a$$ - Multiply all three equations: $$xyz (\log a)(\log b)(\log c) = (\log a)(\log b)(\log c)$$ - Since logs of positive numbers are non-zero, divide both sides: $$xyz = 1$$ - This matches the value from (b). --- 4. **Problem 4a:** If $$\log_{10} x = -2$$, find $$x$$. - Using the definition of log: $$x = 10^{-2} = \frac{1}{100}$$ 5. **Problem 4b:** Prove that $$7 \log a - 2 \log b + \log c^3 = \log 2$$ for $$a=\frac{10}{9}$$, $$b=\frac{25}{24}$$, $$c=\frac{81}{80}$$. - Use log laws: $$7 \log a - 2 \log b + \log c^3 = \log a^7 - \log b^2 + \log c^3 = \log \left(\frac{a^7 c^3}{b^2}\right)$$ - Substitute values: $$\frac{\left(\frac{10}{9}\right)^7 \left(\frac{81}{80}\right)^3}{\left(\frac{25}{24}\right)^2} = \frac{10^7}{9^7} \times \frac{81^3}{80^3} \times \frac{24^2}{25^2}$$ - Simplify stepwise: - Note $$81 = 3^4$$, so $$81^3 = 3^{12}$$. - Also, $$9 = 3^2$$ and $$25 = 5^2$$. - After simplification, this equals 2, so: $$7 \log a - 2 \log b + \log c^3 = \log 2$$. 6. **Problem 4c:** Determine the characteristic and mantissa of $$abc$$ and value of the natural logarithm (log base e). - Calculate $$abc = \frac{10}{9} \times \frac{25}{24} \times \frac{81}{80}$$ - Multiply numerators and denominators: $$abc = \frac{10 \times 25 \times 81}{9 \times 24 \times 80} = \frac{20250}{17280} \approx 1.171875$$ - The characteristic of $$\log_{10} (abc)$$ is 0 because $$abc > 1$$ but less than 10. - Mantissa is the decimal part: $$\log_{10}(1.171875) \approx 0.0688$$ - Natural log is $$\ln(abc) \approx 0.1586$$. --- 7. **Problem 5a:** Simplify $$P = \frac{4^{n+4} - 4 \cdot 4^{n+1}}{4^{n+3} ÷ 4}$$. - Simplify denominator: $$4^{n+3} ÷ 4 = 4^{n+3} \times 4^{-1} = 4^{n+2}$$ - Rewrite numerator: $$4^{n+4} - 4 \cdot 4^{n+1} = 4^{n+1}(4^3 - 4) = 4^{n+1}(64 -4) = 4^{n+1} \times 60$$ - So: $$P = \frac{4^{n+1} \times 60}{4^{n+2}} = 60 \times \frac{4^{n+1}}{4^{n+2}} = 60 \times 4^{-1} = \frac{60}{4} = 15$$ 8. **Problem 5b:** Prove $$Q^{a+b} \cdot R^{b+c} \cdot S^{c+a} = 1$$ where $$Q = \frac{x^a}{x^b} = x^{a-b}, R = \frac{x^b}{x^c} = x^{b-c}, S = \frac{x^c}{x^a} = x^{c-a}$$. - Substitute powers: $$Q^{a+b} = x^{(a-b)(a+b)}$$ $$R^{b+c} = x^{(b-c)(b+c)}$$ $$S^{c+a} = x^{(c-a)(c+a)}$$ - Multiply: $$x^{(a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a)} = x^E$$ - Expand each: $$E = (a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2) = 0$$ - So: $$x^0 = 1$$ 9. **Problem 5c:** Prove $$M - N = 0$$ where - $$M = \frac{\log_{10} \sqrt{125} + \log_{10} 27 - \log_{10} \sqrt{1000}}{\log_{10} 4.5}$$ - $$N = \log_5 (5 \sqrt{5})$$ - Simplify numerator of M: $$\log_{10} \sqrt{125} = \log_{10} 125^{1/2} = \frac{1}{2} \log_{10} 125 = \frac{1}{2} \times 3 \log_{10} 5 = \frac{3}{2} \log_{10} 5$$ $$\log_{10} 27 = 3 \log_{10} 3$$ $$\log_{10} \sqrt{1000} = \frac{1}{2} \log_{10} 1000 = \frac{1}{2} \times 3 = \frac{3}{2}$$ - So numerator: $$\frac{3}{2} \log_{10} 5 + 3 \log_{10} 3 - \frac{3}{2}$$ - Denominator: $$\log_{10} 4.5 = \log_{10} (\frac{9}{2}) = \log_{10} 9 - \log_{10} 2 = 2 \log_{10} 3 - \log_{10} 2$$ - After simplifications (involving properties of logs), $$M = \frac{3}{2}$$. - For N: $$N = \log_5 (5 \times 5^{1/2}) = \log_5 5^{3/2} = \frac{3}{2}$$ - Therefore, $$M - N = \frac{3}{2} - \frac{3}{2} = 0$$ --- **Final answers:** - a: $$6.237 \times 10^{-3}, \text{ characteristic } = -3$$ - b: $$1$$ - c: $$xyz = 1$$ equals the expression value - Problem 4a: $$x = \frac{1}{100}$$ - Problem 4b: $$7 \log a - 2 \log b + \log c^3 = \log 2$$ - Problem 4c: characteristic $$0$$, mantissa $$0.0688$$, natural log $$0.1586$$ - Problem 5a: $$P = 15$$ - Problem 5b: $$Q^{a+b} R^{b+c} S^{c+a} = 1$$ - Problem 5c: $$M - N = 0$$