1. Вычислить: 1) log₂ 16; 2) 4³ᵗᵒ₄ 2; 3) log₄ 8; log₃ 25 4) log₅ 5 5) log₀.² 0,2 - log₀.² 54; 6) log₃ 36- log₃ 12; 7) log₆ 4+ log₆ 9; 8) log₃ 135-log₃ 20 + 2 log₃ 6

Решение:

1. \( \log_2 16 = \log_2 2^4 = 4 \)
2. \( 4^{3 \log_4 2} = (4^{\log_4 2})^3 = 2^3 = 8 \)
3. \( \log_4 8 = \log_{2^2} 2^3 = \frac{3}{2} \log_2 2 = \frac{3}{2} \)
4. \( \frac{\log_3 25}{\log_3 5} = \frac{\log_3 5^2}{\log_3 5} = \frac{2 \log_3 5}{\log_3 5} = 2 \)
5. \( \log_{0.2} 0.2 - \log_{0.2} 54 = \log_{0.2} \frac{0.2}{54} \)
6. \( \log_3 36 - \log_3 12 = \log_3 \frac{36}{12} = \log_3 3 = 1 \)
7. \( \log_6 4 + \log_6 9 = \log_6 (4 \cdot 9) = \log_6 36 = \log_6 6^2 = 2 \)
8. \( \log_3 135 - \log_3 20 + 2 \log_3 6 = \log_3 \frac{135}{20} + \log_3 6^2 = \log_3 \frac{27}{4} + \log_3 36 = \log_3 (\frac{27}{4} \cdot 36) = \log_3 (27 \cdot 9) = \log_3 243 = \log_3 3^5 = 5 \)

Ответ: 1) 4; 2) 8; 3) \(\frac{3}{2}\); 4) 2; 5) \(\log_{0.2} \frac{0.2}{54}\); 6) 1; 7) 2; 8) 5.