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log₁/₃(6-2x) = -1 ⇔
Dziedzina D:
6 - 2x > 0
-2x > -6
x < 3
⇔ 6 - 2x = (-1)¹/³
6 - 2x = ∛(-1)
-2x = -1 - 6
-2x = -7
x = 7/2
Ponieważ x ∉ D - brak rozwiązan
b)
log x + 3log 2 = 3
D: x > 0
log x + log 2³ = 3
log x + log 8 = 3
log (8x) = 3
8x = 10³
x = 1000 / 8
x = 125 ∈ D
c)
log (1-x²) = 1
D: 1 - x² > 0
(1 - x)(1 + x) > 0
x₁ = 1; x₂ = -1
D: x ∈ (-1; 1)
10¹ = 1 - x²
x² + 10 - 1 = 0
x² = -9
Brak rozwiązań
d)
2log₂²x - 5log₂x + 2 = 0
D: x > 0
Wprowadzamy dodatkową zmienną log₂x = t
2t² - 5t + 2 = 0
Δ = (-5)² - 4×2×2 = 25 - 16 = 9
√Δ = 3
t₁ = (5 - 3) / 2×2 = 2/4 = 1/2
t₂ = (5 + 3) / 2×2 = 8/4 = 2
log₂x = t₁ ⇒ log₂x = 1/2 ⇒ x = 2¹/² = √2
log₂x = t₂ ⇒ log₂x = 2 ⇒ x = 2² = 4