Вычислить: 1) \( \log_{\sqrt{3}} \frac{1}{3 \sqrt{3}} \) ; 2) \( \log_{\sqrt{5}} \frac{1}{25 \sqrt[4]{5}} \) ; 3) \( 2^{2-\log_2 5} \) ; 4) \( 3.6^{\log_{3.6} 10 + 1} \)

Решение:

1) \( \log_{\sqrt{3}} \frac{1}{3 \sqrt{3}} = \log_{\sqrt{3}} \frac{1}{3^{1} \cdot 3^{\frac{1}{2}}} = \log_{\sqrt{3}} \frac{1}{3^{\frac{3}{2}}} = \log_{\sqrt{3}} (3^{-\frac{3}{2}}) = \log_{3^{\frac{1}{2}}} (3^{-\frac{3}{2}}) = \frac{-\frac{3}{2}}{\frac{1}{2}} \log_3 3 = -3 \) 2) \( \log_{\sqrt{5}} \frac{1}{25 \sqrt[4]{5}} = \log_{\sqrt{5}} \frac{1}{5^2 \cdot 5^{\frac{1}{4}}} = \log_{\sqrt{5}} \frac{1}{5^{\frac{9}{4}}} = \log_{5^{\frac{1}{2}}} 5^{-\frac{9}{4}} = \frac{-\frac{9}{4}}{\frac{1}{2}} \log_5 5 = -\frac{9}{2} = -4.5 \) 3) \( 2^{2-\log_2 5} = \frac{2^2}{2^{\log_2 5}} = \frac{4}{5} = 0.8 \) 4) \( 3.6^{\log_{3.6} 10 + 1} = 3.6^{\log_{3.6} 10} \cdot 3.6^1 = 10 \cdot 3.6 = 36 \)

Ответ: 1) -3; 2) -4.5; 3) 0.8; 4) 36