384 Вычислить: 1) log √3 3 1; 3 √3 ; 2) log √5 1 ; 25 4√5 ; 3) 22- log 2 5; 4) 3,6 log 3,6 10+1. 5) 2 log5 √5 + 3 log2 8; 6) log2 log2 log2 216.

Вычислим:

1. $$log_{\sqrt{3}} \frac{1}{3\sqrt{3}} = log_{\sqrt{3}} (3\sqrt{3})^{-1} = -log_{\sqrt{3}} (3^{\frac{3}{2}}) = -\frac{3}{2}log_{\sqrt{3}} (\sqrt{3})^2 = -\frac{3}{2} \cdot 2 log_{\sqrt{3}} \sqrt{3} = -3$$
2. $$log_{\sqrt{5}} \frac{1}{25 \sqrt[4]{5}} = log_{\sqrt{5}} (5^2 \cdot 5^{\frac{1}{4}})^{-1} = log_{\sqrt{5}} (5^{\frac{9}{4}})^{-1} = log_{\sqrt{5}} (\sqrt{5})^{\frac{-9}{2}} = -\frac{9}{2}log_{\sqrt{5}} \sqrt{5} = -\frac{9}{2} = -4.5$$
3. $$2^{2 - log_2 5} = 2^2 \cdot 2^{-log_2 5} = 4 \cdot \frac{1}{2^{log_2 5}} = 4 \cdot \frac{1}{5} = \frac{4}{5} = 0.8$$
4. $$3.6^{log_{3.6} 10 + 1} = 3.6^{log_{3.6} 10} \cdot 3.6^1 = 10 \cdot 3.6 = 36$$
5. $$2 log_5 \sqrt{5} + 3 log_2 8 = 2 log_5 5^{\frac{1}{2}} + 3 log_2 2^3 = 2 \cdot \frac{1}{2} log_5 5 + 3 \cdot 3 log_2 2 = 1 + 9 = 10$$
6. $$log_2 log_2 log_2 2^{16} = log_2 log_2 (16 log_2 2) = log_2 log_2 16 = log_2 log_2 2^4 = log_2 (4 log_2 2) = log_2 4 = log_2 2^2 = 2$$

Ответ: 1) -3; 2) -4.5; 3) 0.8; 4) 36; 5) 10; 6) 2