Задача 79155 ...
Условие
1) log₂√2;
2) log₃√3;
3) log₅√5;
4) log₃ 1/√3;
5) log₇ 1/√7;
6) lg √10;
7) lg √10;
8) ln √e;
9) ln √e;
10) log₁₁ √11;
11) log₅ 0,2;
12) log₃ √3/2;
13) log₃ √3/3;
14) log₅ √5/5;
15) lg √100.
Решение
Самое главное, нужно запомнить две формулы:
1) [m]\large \log_{a} a = 1[/m]
2) [m]\large \log_{a} b^{c} = c \cdot \log_{a} b[/m]
Решаем
1) [m]\large \log_{2} \sqrt{2} = \log_{2} 2^{1/2} = \frac{1}{2} \cdot \log_{2} 2 = \frac{1}{2}[/m]
2) [m]\large \log_{3} \sqrt{3} = \log_{3} 3^{1/2} = \frac{1}{2} \cdot \log_{3} 3 = \frac{1}{2}[/m]
3) [m]\large \log_{5} \sqrt[3]{5} = \log_{5} 5^{1/3} = \frac{1}{3} \cdot \log_{5} 5 = \frac{1}{3}[/m]
4) [m]\large \log_{3} \frac{1}{\sqrt{3}} = \log_{3} 3^{-1/2} = -\frac{1}{2} \cdot \log_{3} 3 = -\frac{1}{2}[/m]
5) [m]\large \log_{7} \frac{1}{\sqrt{7}} = \log_{7} 7^{-1/2} = -\frac{1}{2} \cdot \log_{7} 7 = -\frac{1}{2}[/m]
6) [m]\large \lg \sqrt{10} = \lg 10^{1/2} = \frac{1}{2} \cdot \lg 10 = \frac{1}{2}[/m]
7) [m]\large \lg \sqrt[3]{10} = \lg 10^{1/3} = \frac{1}{3} \cdot \lg 10 = \frac{1}{3}[/m]
8) [m]\large \ln \sqrt{e} = \ln e^{1/2} = \frac{1}{2} \cdot \ln e = \frac{1}{2}[/m]
9) [m]\large \ln \sqrt[4]{e} = \ln e^{1/4} = \frac{1}{4} \cdot \ln e = \frac{1}{4}[/m]
10) [m]\large \log_{11} \sqrt[5]{11} = \log_{11} 11^{1/5} = \frac{1}{5} \cdot \log_{11} 11 = \frac{1}{5}[/m]
11) [m]\large \log_{1/5} \sqrt{0,2} = \log_{1/5} (\frac{1}{5})^{1/2} = \frac{1}{2} \cdot \log_{1/5} \frac{1}{5} = \frac{1}{2}[/m]
12) [m]\large \log_{1/5} \sqrt[3]{0,2} = \log_{1/5} (\frac{1}{5})^{1/3} = \frac{1}{3} \cdot \log_{1/5} \frac{1}{5} = \frac{1}{3}[/m]
13) [m]\large \log_{3} \sqrt[7]{3} = \log_{3} 3^{1/7} = \frac{1}{7} \cdot \log_{3} 3 = \frac{1}{7}[/m]
14) [m]\large \log_{5} \sqrt[5]{5} = \log_{5} 5^{1/5} = \frac{1}{5} \cdot \log_{5} 5 = \frac{1}{5}[/m]
15) [m]\large \lg \sqrt[5]{100} = \lg (10^2)^{1/5} = \lg (10)^{2/5} = \frac{2}{5} \cdot \lg 10 = \frac{2}{5}[/m]