与えられた20個の対数計算の問題を解き、答えを有理数で求める。

1. 問題の内容

与えられた20個の対数計算の問題を解き、答えを有理数で求める。

2. 解き方の手順

(1) $\log_7 7 = 1$ (底と真数が等しいので)

(2) $\log_\pi 1 = 0$ (どんな底でも、真数が1なら0)

(3) $3^{2\log_3 2} = 3^{\log_3 2^2} = 3^{\log_3 4} = 4$

(4) $\log_{\sqrt{5}} x = 4$ より $x = (\sqrt{5})^4 = (5^{1/2})^4 = 5^2 = 25$

(5) $\log_{\sqrt{2}} 8 = \log_{2^{1/2}} 2^3 = \frac{3}{1/2} = 6$

(6) $\log_3 \frac{1}{9} = \log_3 3^{-2} = -2$

(7) $\log_{0.5} 0.25 = \log_{\frac{1}{2}} \frac{1}{4} = \log_{2^{-1}} 2^{-2} = \frac{-2}{-1} = 2$

(8) $\log_{10} 10\sqrt{10} = \log_{10} (10 \cdot 10^{1/2}) = \log_{10} 10^{3/2} = \frac{3}{2}$

(9) $\log_6 2 + \log_6 18 = \log_6 (2 \cdot 18) = \log_6 36 = \log_6 6^2 = 2$

(10) $\log_2 48 - \log_2 6 = \log_2 \frac{48}{6} = \log_2 8 = \log_2 2^3 = 3$

(11) $\log_{10} 2 - \log_{10} 200 = \log_{10} \frac{2}{200} = \log_{10} \frac{1}{100} = \log_{10} 10^{-2} = -2$

(12) $\log_3 \frac{45}{4} + \log_3 \frac{12}{5} = \log_3 (\frac{45}{4} \cdot \frac{12}{5}) = \log_3 (9 \cdot 3) = \log_3 27 = \log_3 3^3 = 3$

(13) $\log_2 \frac{2}{3} - \log_2 \frac{1}{12} = \log_2 (\frac{2}{3} \div \frac{1}{12}) = \log_2 (\frac{2}{3} \cdot 12) = \log_2 8 = \log_2 2^3 = 3$

(14) $2\log_5 \frac{25}{2} - \frac{2}{3} \log_5 \frac{125}{8} = \log_5 (\frac{25}{2})^2 - \log_5 (\frac{125}{8})^{2/3} = \log_5 \frac{625}{4} - \log_5 (\frac{25}{4}) = \log_5 (\frac{625}{4} \div \frac{25}{4}) = \log_5 25 = \log_5 5^2 = 2$

(15) $\log_{10} 2 + \log_{10} \sqrt{15} - \frac{1}{2} \log_{10} \frac{3}{5} = \log_{10} (2 \cdot \sqrt{15}) - \log_{10} (\frac{3}{5})^{1/2} = \log_{10} \frac{2\sqrt{15}}{\sqrt{\frac{3}{5}}} = \log_{10} \frac{2\sqrt{15}}{\frac{\sqrt{3}}{\sqrt{5}}} = \log_{10} \frac{2\sqrt{15}\sqrt{5}}{\sqrt{3}} = \log_{10} (2\sqrt{5}\sqrt{5}) = \log_{10} (2 \cdot 5) = \log_{10} 10 = 1$

(16) $\log_2 6 - \log_4 9 = \log_2 6 - \log_{2^2} 3^2 = \log_2 6 - \log_2 3 = \log_2 \frac{6}{3} = \log_2 2 = 1$

(17) $\log_8 16 = \log_{2^3} 2^4 = \frac{4}{3}$

(18) $\log_8 4 + \log_{16} 8 = \log_{2^3} 2^2 + \log_{2^4} 2^3 = \frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$

(19) $(\log_2 5)(\log_5 8) = (\log_2 5)(\log_5 2^3) = 3 (\log_2 5)(\log_5 2) = 3 (\log_2 5) \cdot \frac{1}{\log_2 5} = 3$

(20) $(\log_2 5 + \log_4 \frac{1}{5})(\log_5 2 + \log_{25} \frac{1}{2}) = (\log_2 5 + \frac{\log_2 \frac{1}{5}}{\log_2 4})(\log_5 2 + \frac{\log_5 \frac{1}{2}}{\log_5 25}) = (\log_2 5 + \frac{-\log_2 5}{2})(\log_5 2 + \frac{-\log_5 2}{2}) = (\log_2 5 - \frac{1}{2}\log_2 5)(\log_5 2 - \frac{1}{2}\log_5 2) = (\frac{1}{2}\log_2 5)(\frac{1}{2}\log_5 2) = \frac{1}{4} (\log_2 5) (\log_5 2) = \frac{1}{4} (\log_2 5) (\frac{1}{\log_2 5}) = \frac{1}{4}$

3. 最終的な答え

(1) 1

(2) 0

(3) 4

(4) 25

(5) 6

(6) -2

(7) 2

(8) 3/2

(9) 2

(10) 3

(11) -2

(12) 3

(13) 3

(14) 2

(15) 1

(16) 1

(17) 4/3

(18) 17/12

(19) 3

(20) 1/4