Краткое решение задач с логарифмами

Вот решения задач с первой страницы в кратком виде, удобном для записи в тетрадь. 1. \(\log_3 1,8 + \log_3 5 = \log_3 (1,8 \cdot 5) = \log_3 9 = 2\) 2. \((\log_2 16) \cdot (\log_6 36) = 4 \cdot 2 = 8\) 3. \(7 \cdot 5^{\log_5 4} = 7 \cdot 4 = 28\) 4. \(36^{\log_6 5} = (6^2)^{\log_6 5} = 6^{2\log_6 5} = 6^{\log_6 5^2} = 5^2 = 25\) 5. \(\log_{0,25} 2 = \log_{2^{-2}} 2 = -\frac{1}{2} \cdot \log_2 2 = -0,5\) 6. \(\log_4 8 = \log_{2^2} 2^3 = \frac{3}{2} \cdot \log_2 2 = 1,5\) 7. \(\log_5 60 - \log_5 12 = \log_5 \frac{60}{12} = \log_5 5 = 1\) 8. \(\log_{0,2} 2 + \log_{0,2} 5 = \log_{0,2} (2 \cdot 5) = \log_{0,2} 10 = \log_{10^{-1}} 10 = -1\) 9. \(\log_{0,3} 10 - \log_{0,3} 3 = \log_{0,3} \frac{10}{3} = \log_{\frac{3}{10}} \frac{10}{3} = -1\) 10. \(\frac{\log_3 25}{\log_3 5} = \log_5 25 = 2\) 11. \(\frac{\log_7 13}{\log_{49} 13} = \frac{\log_7 13}{\frac{1}{2}\log_7 13} = 2\) 12. \(\log_5 9 \cdot \log_3 25 = \log_5 3^2 \cdot \log_3 5^2 = 2\log_5 3 \cdot 2\log_3 5 = 4 \cdot \log_5 3 \cdot \frac{1}{\log_5 3} = 4\) 13. \(\frac{9^{\log_5 50}}{9^{\log_5 2}} = 9^{\log_5 50 - \log_5 2} = 9^{\log_5 \frac{50}{2}} = 9^{\log_5 25} = 9^2 = 81\) 14. \((1 - \log_2 12)(1 - \log_6 12) = (1 - (\log_2 4 + \log_2 3))(1 - (\log_6 6 + \log_6 2)) = \) \(= (1 - 2 - \log_2 3)(1 - 1 - \log_6 2) = (-1 - \log_2 3)(-\log_6 2) = (1 + \log_2 3) \cdot \frac{1}{\log_2 6} = \) \(= \log_2 (2 \cdot 3) \cdot \frac{1}{\log_2 6} = \log_2 6 \cdot \frac{1}{\log_2 6} = 1\) Решения со второй страницы: 15. \(6\log_7 \sqrt[3]{7} = 6 \cdot \frac{1}{3} \log_7 7 = 2\) 16. \(\log_{\sqrt[13]{13}} 13 = \log_{13^{1/13}} 13 = 13 \cdot \log_{13} 13 = 13\) 17. \(\frac{\log_3 18}{2 + \log_3 2} = \frac{\log_3 18}{\log_3 9 + \log_3 2} = \frac{\log_3 18}{\log_3 18} = 1\) 18. \(\frac{\log_3 5}{\log_3 7} + \log_7 0,2 = \log_7 5 + \log_7 0,2 = \log_7 (5 \cdot 0,2) = \log_7 1 = 0\) 19. \(\log_{0,8} 3 \cdot \log_3 1,25 = \log_{0,8} 1,25 = \log_{4/5} \frac{5}{4} = -1\) 20. \(5^{\log_{25} 49} = 5^{\log_{5^2} 7^2} = 5^{\frac{2}{2}\log_5 7} = 5^{\log_5 7} = 7\) 21. \(\log^2_{\sqrt{7}} 49 = (\log_{7^{1/2}} 7^2)^2 = (2 \cdot 2 \log_7 7)^2 = 4^2 = 16\) 22. \(5^{3 + \log_5 2} = 5^3 \cdot 5^{\log_5 2} = 125 \cdot 2 = 250\) 23. \(8^{2\log_8 3} = 8^{\log_8 3^2} = 3^2 = 9\) 24. \(64^{\log_8 \sqrt{3}} = (8^2)^{\log_8 \sqrt{3}} = 8^{2\log_8 \sqrt{3}} = 8^{\log_8 (\sqrt{3})^2} = 3\) 25. \(\log_4 \log_5 25 = \log_4 2 = \log_{2^2} 2 = 0,5\) 26. \(\frac{24}{3^{\log_3 2}} = \frac{24}{2} = 12\) 27. \(\log_{\frac{1}{13}} \sqrt{13} = \log_{13^{-1}} 13^{1/2} = \frac{1/2}{-1} = -0,5\) 28. \(\log_3 8,1 + \log_3 10 = \log_3 (8,1 \cdot 10) = \log_3 81 = 4\) 29. \(\frac{\log_6 \sqrt{13}}{\log_6 13} = \frac{\frac{1}{2}\log_6 13}{\log_6 13} = 0,5\)