Question

Логи. Матеша. Срочно. 100 балов. Задание на фото.

Answer 1

$1. ~ 1) ~ \log_{8}8 = 1$

$2) ~ \log_{8}\dfrac{1}{64} = \log_{8}\dfrac{1}{8^{2}} = \log_{8}8^{-2} = -2$

$3) ~ \log_{8}8\sqrt[5]{8} = \log_{8}\left(8^{1} \cdot 8^{\tfrac{1}{5}}\right) = \log_{8}8^{1\tfrac{1}{5} } = 1\dfrac{1}{5}$

$4) ~ \log_{8}8^{-3} = -3$

$2. ~ 1) ~ \log_{2}48 + \log_{2}\dfrac{1}{3} = \log_{2}\left(48 \cdot \dfrac{1}{3} \right) = \log_{2}16 = 4$

$2) ~ \lg 40 + 2\lg 5 =\lg 40 + \lg 5^{2} = \lg 40 + \lg 25 = \lg (40 \cdot 25) = \lg 1000 = 3$

$3) ~ \log_{2}\sqrt{12} - \log_{2}\sqrt{6} = \log_{2} \dfrac{\sqrt{12}}{\sqrt{6}} =\log_{2} \sqrt{\dfrac{12}{6} } = \log_{2}\sqrt{2} = \dfrac{1}{2}$

$4) ~ \pi^{2\log_{\pi}7} = \pi^{\log_{\pi}7^{2}} = \pi^{\log_{\pi}49} = 49$

$3. ~ 1) ~ 3^{\tfrac{2}{\log_{7}3} } = 3^{2 \log_{3}7} = 3^{\log_{3}7^{2}} = 3^{\log_{3}49} = 49$

$2) ~ 25^{1 + \log_{5}3} = 25^{1} \cdot 25^{\log_{5}3} = 25 \cdot 5^{2\log_{5}3} = 25 \cdot 5^{\log_{5}9} = 25 \cdot 9 = 225$

$3) ~ \dfrac{\lg 16}{\lg 2} = \dfrac{\lg 2^{4}}{\lg 2} = \dfrac{4\lg 2}{\lg 2} = 4$

Использованные свойства и следствия логарифмов:

$1) ~ \log_{a}(xy) = \log_{a}x + \log_{a}y ~ ~~ (a > 0, ~ a \neq 1, ~ x > 0, ~ y > 0)$

$2) ~ \log_{a} \left(\dfrac{x}{y} \right) = \log_{a}x - \log_{a}y ~~~ (a > 0, ~ a \neq 1, ~ x > 0, ~ y > 0)$

$3) ~ a^{\log_{a}b} = b ~~~ (a > 0, ~ a \neq 1, ~b > 0)$

$4) ~ \log_{a}x^{p} = p \log_{a}x ~ (a > 0, ~ a \neq 1, ~ x > 0)$

$5) ~ \log_{a}b = \dfrac{1}{\log_{b}a} ~~~ (a>0, ~ a\neq 1, ~ b > 0, ~b\neq 1)$