Найдите произведение уравнения log² 0.2 (x/25) + log² 0.2 (x/5) = 1

1) log₀₎₂² (x/25) = log₀₎₂ (x/25) * log₀₎₂ (x/25) =

= (log ₀₎₂x - log₀₎₂25) * (log ₀₎₂x - log₀₎₂25) = (log₀₎₂x + 2) * (log ₀₎₂x + 2) =

= (log ₀₎₂x + 2) ² = log₀₎₂²x + 4log₀₎₂x + 4

2) log₀₎₂² (x/5) = log₀₎₂ (x/5) * log₀₎₂ (x/5) = (log₀₎₂x - log₀₎₂5) * (log ₀₎₂x - log₀₎₂5) =

= (log₀₎₂x + 1) * (log₀₎₂x + 1) = (log₀₎₂x + 1) ² = log₀₎₂²x + 2log₀₎₂x + 1

3) Само уравнение:

log₀₎₂²x + 4log₀₎₂x + 4 + log₀₎₂²x + 2log₀₎₂x + 1 = 1 (ОДЗ: x > 0)

log₀₎₂x = t

t² + 4t + 4 + t² + 2t = 0

2t² + 6t + 4 = 0

t² + 3t + 2 = 0

По т. Виета

а) t = - 2, ⇒ log₀₎₂x = - 2, x = 0,2⁻² = 25

б) t = - 1, ⇒ log₀₎₂x = - 1, ⇒ x = 0,2⁻¹ = 5

Ответ: 125