第10章 10.12

(1)

$\log 2^{20} = 20\log 2 = 20 \cdot 0.3010 = 6.02$

$\log 3^{14} = 14\log 3 = 14 \cdot 0.4771 = 6.6794$

これから$3^{14}$が大きい

(2)

$\log 6^{200} = 200(\log 2 + \log 3) = 200(0.3010 + 0.4771) = 155.62$

$\log 10^{155} < 155 < \log 10^{156}$

よって156桁

(3)

$\log 2^{31} = 31 \cdot 0.3010 = 9.331$

$\log 10^{9} < 9.331 < \log 10^{10}$

よって10桁

(4)

$\log\left(\frac{81}{80}\right)^{2000} = 2000 \cdot (\log 3^4 - \log 2^3 - \log 10) = 2000 \cdot (4 \cdot 0.4771 - 3 \cdot 0.3010 - 1) = 10.8$

$\log 10^{10} < 10.8 < \log 10^{11}$

よって11桁

(5)

$\log\left(\frac{1}{1.25}\right)^{200} = 200 \cdot (-3\log 5 - \log 0.01) = -200(-3\log 10 + 3\log 2 - \log 0.01)$

$= -200(-3 + 0.903 + 2) = -19.4$

$\log 10^{-20} < -19.4 < \log 10^{-19}$

よって20桁

(6)

$\log\frac{5}{4} = \log\frac{10}{8} = \log 10 - 3\log 2 = 1 - 3 \cdot 0.3010 = 0.097$

$\log 4000 < n \cdot \log\frac{5}{4} < \log 5000 \qquad \log 4 + \log 1000 < n \cdot 0.097 < \log 5 + \log 1000$

$2\log 2 + 3 < n \cdot 0.097 < \log 10 - \log 2 + 3 \qquad 3.602 < n \cdot 0.097 < 3.699$

$\frac{3.602}{0.097} < n < \frac{3.699}{0.097} \qquad 37.134 < n < 38.134 \qquad \text{これから } n = 38$

(7)

$\log 10^3 < \log 2.7^n < \log 10^4 \qquad 3 < n \cdot \log 2.7 < 4$

$\log 2.7 = \log 27 - \log 10 = 3\log 3 - 1 = 0.4313$

$\frac{3}{0.4313} < n < \frac{4}{0.4313} \qquad 6.9557 < n < 9.274$

これから7,8,9