与えられた6つの対数の式をそれぞれ計算します。 (1) $\log_2 8 + \log_2 \frac{1}{\sqrt{2}} - \log_2 4$ (2) $\log_3 \sqrt{2}$ (3) $\frac{1}{2} \log_2 3 + \log_4 28 - 3 \log_8 \sqrt{21}$ (4) $\log_3 54 - \frac{1}{3} \log_3 162 - \frac{1}{3} \log_3 4$ (5) $\log_3 5 \cdot \log_5 3$ (6) $\log_2 3 \cdot \log_{27} 25 \cdot \log_5 32$

1. 問題の内容

与えられた6つの対数の式をそれぞれ計算します。

(1) $\log_2 8 + \log_2 \frac{1}{\sqrt{2}} - \log_2 4$

(2) $\log_3 \sqrt{2}$

(3) $\frac{1}{2} \log_2 3 + \log_4 28 - 3 \log_8 \sqrt{21}$

(4) $\log_3 54 - \frac{1}{3} \log_3 162 - \frac{1}{3} \log_3 4$

(5) $\log_3 5 \cdot \log_5 3$

(6) $\log_2 3 \cdot \log_{27} 25 \cdot \log_5 32$

2. 解き方の手順

(1)

$\log_2 8 + \log_2 \frac{1}{\sqrt{2}} - \log_2 4 = \log_2 2^3 + \log_2 2^{-\frac{1}{2}} - \log_2 2^2 = 3 - \frac{1}{2} - 2 = \frac{1}{2}$

(2)

$\log_3 \sqrt{2} = \log_3 2^{\frac{1}{2}} = \frac{1}{2} \log_3 2$

(3)

$\frac{1}{2} \log_2 3 + \log_4 28 - 3 \log_8 \sqrt{21} = \frac{1}{2} \log_2 3 + \frac{\log_2 28}{\log_2 4} - 3 \frac{\log_2 \sqrt{21}}{\log_2 8} = \frac{1}{2} \log_2 3 + \frac{\log_2 28}{2} - 3 \frac{\log_2 21^{\frac{1}{2}}}{3} = \frac{1}{2} \log_2 3 + \frac{1}{2} \log_2 28 - \frac{1}{2} \log_2 21 = \frac{1}{2} (\log_2 3 + \log_2 28 - \log_2 21) = \frac{1}{2} \log_2 \frac{3 \cdot 28}{21} = \frac{1}{2} \log_2 4 = \frac{1}{2} \cdot 2 = 1$

(4)

$\log_3 54 - \frac{1}{3} \log_3 162 - \frac{1}{3} \log_3 4 = \log_3 54 - \frac{1}{3} (\log_3 162 + \log_3 4) = \log_3 54 - \frac{1}{3} \log_3 (162 \cdot 4) = \log_3 54 - \frac{1}{3} \log_3 648 = \log_3 54 - \log_3 \sqrt[3]{648} = \log_3 \frac{54}{\sqrt[3]{648}} = \log_3 \frac{54}{\sqrt[3]{216 \cdot 3}} = \log_3 \frac{54}{6 \sqrt[3]{3}} = \log_3 \frac{9}{\sqrt[3]{3}} = \log_3 \frac{3^2}{3^{\frac{1}{3}}} = \log_3 3^{2-\frac{1}{3}} = \log_3 3^{\frac{5}{3}} = \frac{5}{3}$

(5)

$\log_3 5 \cdot \log_5 3 = \log_3 5 \cdot \frac{\log_3 3}{\log_3 5} = \log_3 5 \cdot \frac{1}{\log_3 5} = 1$

(6)

$\log_2 3 \cdot \log_{27} 25 \cdot \log_5 32 = \log_2 3 \cdot \frac{\log_5 25}{\log_5 27} \cdot \log_5 32 = \log_2 3 \cdot \frac{2}{3 \log_5 3} \cdot 5 \log_5 2 = \log_2 3 \cdot \frac{2}{3} \cdot \frac{1}{\log_5 3} \cdot 5 \log_5 2 = \log_2 3 \cdot \frac{2}{3} \cdot \log_3 5 \cdot 5 \log_5 2 = \frac{10}{3} \log_2 3 \cdot \log_3 5 \cdot \log_5 2 = \frac{10}{3} \frac{\log_3 3}{\log_3 2} \cdot \log_3 5 \cdot \log_5 2 = \frac{10}{3} \frac{1}{\log_3 2} \cdot \log_3 5 \cdot \frac{\log_3 2}{\log_3 5} = \frac{10}{3} \frac{1}{\log_3 2} \cdot \frac{\log_3 2 \cdot \log_3 5}{\log_3 5} = \frac{10}{3}$

3. 最終的な答え

(1) $\frac{1}{2}$

(2) $\frac{1}{2} \log_3 2$

(3) $1$

(4) $\frac{5}{3}$

(5) $1$

(6) $\frac{10}{3}$