Question

решите пожалуйста уравнение​

Answer 1

1.

log₄(log₃x) = 1

log₃x = 4¹

x = 3⁴ = 81

2.

lgx - lg5 = 2lgx - lg(9 - x)

lgx - 2lgx + lg(9 - x) = lg5

-lgx + lg(9 - x) = lg5

lg((9 - x) / x) = lg5

(9 - x) / x = 5

x = 1.5

3.

lgx = t

t² - t = 6

t₁ = -2  |  t₂ = 3

lgx₁ = -2

x₁ = 1 / e²

x₂ = e³

4.

$\frac{log_{10}16}{log_{10}x^2} + \frac{log_{10}64}{log_{10}2x} = 3\\\frac{log_{10}2^4}{log_{10}x^2} + \frac{log_{10}2^6}{log_{10}2x} = 3\\\frac{4log_{10}2}{log_{10}x^2} + \frac{6log_{10}2}{log_{10}2x} = 3\\\frac{4log_{10}(2)log_{10}(2x) + 6log_{10}(2)log_{10}(x^2)}{log_{10}(x^2)log_{10}(2x)} = 3\\4log_{10}(2)log_{10}(2x) + 6log_{10}(2)log_{10}(x^2) = 3log_{10}(x^2)log_{10}(2x)\\4log_{10}(2)(log_{10}(2) + log_{10}(x)) + 12log_{10}(2)log_{10}(|x|) = 6log_{10}(|x|)log_{10}(2x)$

$4log^2_{10}(2) + 4log_{10}(2)log_{10}(x)+ 12log_{10}(2)log_{10}(x) = 6log_{10}(x)log_{10}(2x)\\-3log^2_{10}(x) + 5log_{10}(2)log_{10}(x) + 2log^2_{10}(2) = 0\\t = log_{10}(x)\\-3t^2 + 5log_{10}(2)t + 2log^2_{10}(2) = 0\\t_1 = 2log_{10}(2)\\t_2 = -\frac{log_{10}(2)}{3}\\log_{10}(x_1) = 2log_{10}(2)\\x_1 = 4\\log_{10}(x_2) = -\frac{log_{10}(2)}{3}\\x_2 = \frac{\sqrt[3]{4}}{2}$

5.

$log_{3^{-1}}(x^{log_{3^{-1}}(x)-\frac{2}{3}}) = log_{3^{-1}}(3^{\frac{8}{3}})\\log_{3^{-1}}(x^{-log_{3}(x)-\frac{2}{3}}) = -\frac{8}{3}\\(log_3(x) + \frac{2}{3})log_3(x) = -\frac{8}{3}\\log^2_3(x) + \frac{2}{3}log_3(x) = -\frac{8}{3}\\t = log_3(x)\\t^2 + \frac{2}{3}t = -\frac{8}{3}\\t \notin R\\x \notin R$