766. 1) log2 √2; 2) log3 (1 / (3√3)); 3) log0,5 (1 / √32); 4) log7 (√7 / 49); 5) log128 64; 6) log27 243; 7) log64 256; 8) log81 27; 9) log √3 (1 / (3 * 3√3)); 10) log1/2 (1 / (4√2))

Решение:

1) \( \log_2 \sqrt{2} = \log_2 2^{1/2} = \frac{1}{2} \)

2) \( \log_3 \frac{1}{3\sqrt{3}} = \log_3 \frac{1}{3^{1} \cdot 3^{1/2}} = \log_3 \frac{1}{3^{3/2}} = \log_3 3^{-3/2} = -\frac{3}{2} \)

3) \( \log_{0.5} \frac{1}{\sqrt{32}} = \log_{1/2} \frac{1}{2^{5/2}} = \log_{1/2} 2^{-5/2} = \log_{1/2} (1/2)^{5/2} = \frac{5}{2} \)

4) \( \log_7 \frac{\sqrt{7}}{49} = \log_7 \frac{7^{1/2}}{7^2} = \log_7 7^{1/2 - 2} = \log_7 7^{-3/2} = -\frac{3}{2} \)

5) \( \log_{128} 64 = \log_{2^7} 2^6 = \frac{6}{7} \)

6) \( \log_{27} 243 = \log_{3^3} 3^5 = \frac{5}{3} \)

7) \( \log_{64} 256 = \log_{2^6} 2^8 = \frac{8}{6} = \frac{4}{3} \)

8) \( \log_{81} 27 = \log_{3^4} 3^3 = \frac{3}{4} \)

9) \( \log_{\sqrt{3}} \frac{1}{3\sqrt[3]{3}} = \log_{3^{1/2}} \frac{1}{3^{1 + 1/3}} = \log_{3^{1/2}} \frac{1}{3^{4/3}} = \log_{3^{1/2}} 3^{-4/3} = \frac{-4/3}{1/2} = -\frac{4}{3} \cdot 2 = -\frac{8}{3} \)

10) \( \log_{1/2} \frac{1}{4\sqrt{2}} = \log_{2^{-1}} \frac{1}{2^2 \cdot 2^{1/2}} = \log_{2^{-1}} \frac{1}{2^{5/2}} = \log_{2^{-1}} 2^{-5/2} = \frac{-5/2}{-1} = \frac{5}{2} \)

Ответ:

• 1) \( \frac{1}{2} \)
• 2) \( -\frac{3}{2} \)
• 3) \( \frac{5}{2} \)
• 4) \( -\frac{3}{2} \)
• 5) \( \frac{6}{7} \)
• 6) \( \frac{5}{3} \)
• 7) \( \frac{4}{3} \)
• 8) \( \frac{3}{4} \)
• 9) \( -\frac{8}{3} \)
• 10) \( \frac{5}{2} \)