Задача 77757 Решить уравнение (log_5 5/x^2)/(log_5...

5f4e19cc6a894c0287cf31f4

6/9/2024 11:48:03 AM

Решить уравнение (log_5 5/x^2)/(log_5 X)^2=3

626fab9a354ab65fb2f43abb

6/9/2024 3:57:57 PM

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[m]\frac{\log_5 (5/x^2)}{(\log_5 x)^2} = 3[/m]
Область определения:
{ x > 0
{ log_5 x ≠ 0
Получаем:
{ x > 0
{ x ≠ 1
x ∈ (0; 1) U (1; +oo)
[m]\frac{\log_5 5 - \log_5 x^2}{(\log_5 x)^2} = 3[/m]
[m]\frac{1 - 2\log_5 x}{(\log_5 x)^2} = 3[/m]
Замена t = log_5 x
[m]\frac{1 - 2t}{t^2} = 3[/m]
[m]0 = 3 - \frac{1 - 2t}{t^2}[/m]
[m]\frac{3t^2 + 2t - 1}{t^2} = 0[/m]
3t^2 + 2t - 1 = 0
(t + 1)(3t - 1) = 0
[m]t_1 = \log_5 x = -1[/m]
[m]x_1 = 5^{-1} = \frac{1}{5}[/m]
[m]t_2 = log_5 x = \frac{1}{3}[/m]
[m]x_2 = 5^{1/3} = \sqrt[3]{5}[/m]