Решите уравнение 90 — 19.3x−1 + 2 = 0.

Решение:

Решим уравнение 9^x − 19 ⋅ 3^{x − 1} + 2 = 0.

Преобразуем уравнение:

9^x - 19 ⋅ 3^x ⋅ 3^-1 + 2 = 0

9^x - \(\frac{19}{3}\) ⋅ 3^x + 2 = 0

Пусть t = 3^x, тогда t > 0.

(3^x)² - \(\frac{19}{3}\) ⋅ 3^x + 2 = 0

t² - \(\frac{19}{3}\) ⋅ t + 2 = 0

Умножим обе части уравнения на 3:

3t² - 19t + 6 = 0

Решим квадратное уравнение.

D = b² - 4ac = (-19)² - 4 ⋅ 3 ⋅ 6 = 361 - 72 = 289

t_1,2 = \(\frac{-b ± \\sqrt{D}}{2a}\) = \(\frac{19 ± \\sqrt{289}}{2 ⋅ 3}\) = \(\frac{19 ± 17}{6}\)

t₁ = \(\frac{19 + 17}{6}\) = \(\frac{36}{6}\) = 6

t₂ = \(\frac{19 - 17}{6}\) = \(\frac{2}{6}\) = \(\frac{1}{3}\)

Сделаем обратную замену:

3^x = 6

x = log₃6

3^x = \(\frac{1}{3}\)

3^x = 3^-1

x = -1

Ответ: -1; log₃6