Odpowiedź:
c)
4 * 16ˣ = 4ˣ⁺¹ - 1
4 * (4²)ˣ = 4ˣ⁺¹ - 1
4 * 4²ˣ = 4ˣ * 4 - 1
za 4ˣ wstawiamy z
4z² = 4z - 1
4z² - 4z + 1 = 0
Δ = (- 4)² - 4 * 4 * 1 = 16 - 16 = 0
z₁ = z₂ = 4/8 = 1/2
4ˣ = 1/2
(2²)x = 2⁻¹
2²ˣ = 2⁻¹
2x = - 1
x = - 1/2
d)
4ˣ + 8 = 3 * 2ˣ⁺¹
(2²)ˣ + 8 = 3 * 2ˣ * 2
2²ˣ + 8 =6 * 2ˣ
za 2ˣ wstawiamy z
z² + 8 = 6z
x² - 6z + 8 = 0
Δ = (- 6)² - 4 * 1 * 8 = 36 - 32 = 4
√Δ = √4 = 2
z₁ = ( 6 - 2)/2 = 4/2 = 2
z₂ = (6 + 2)/2 = 8/2 = 4
2ˣ = 2 ∨ 2ˣ = 4
2ˣ = 2¹ ∨ 2ˣ = 2²
x = 1 ∨ x = 2
v - znaczy lub
e)
1/4ˣ - 3/2ˣ + 2 = 0
1/(2²)ˣ - 3/2ˣ +2 = 0
1/2²ˣ - 3/2ˣ + 2 = 0
za 2ˣ wstawiamy z
1/z² - 3/z + 2 = 0 | * z²
1 - 3z + 2z² = 0
2z² - 3z + 1 = 0
Δ = (- 3)² - 4 * 2 * 1 = 9 - 8 = 1
√Δ = √1 = 1
z₁ = (3 - 1)/4 = 2/4 = 1/2
z₂ = (3 + 1)/4 = 1
2ˣ = 1/2 ∨ 2ˣ = 1
2ˣ = 2⁻¹ ∨ 2ˣ = 2⁰
x = - 1 ∨ x = 0
