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sammy8
@sammy8
July 2022
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Помогите решить, подробно, пожалуйста
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Dимасuk
Verified answer
ОДЗ:
25ˣ⁺³ - 1> 0
5ˣ⁺³ + 1> 0
25ˣ⁺³ > 25⁰
5ˣ⁺³ > -1 ⇒ x ∈ R
x + 3 > 0
x > -3
log₂(25ˣ⁺³ - 1) = 2 + log₂(5ˣ⁺³ + 1)
log₂(25ˣ⁺³ - 1) = log₂4 + log₂(5ˣ⁺³ + 1)
log₂(25ˣ⁺³ - 1) = log₂(4·5ˣ⁺³ + 4)
25ˣ⁺³ - 1 = 4·5ˣ⁺³ + 4
25ˣ⁺³ - 4·5ˣ⁺³ - 4 - 1 = 0
25ˣ⁺³ - 4·5ˣ⁺³ - 5 = 0
Пусть t = 5ˣ⁺³, t > 0
t² - 4t - 5 = 0
t² - 4t + 4 - 9 = 0
(t - 2)² - 3² = 0
(t - 2 - 3)(t - 2 + 3) = 0
(t - 5)(t + 1) = 0
t = 5; t = -1 - не подходит
Обратная замена:
5ˣ⁺³ = 5
5ˣ⁺³ = 5¹
x + 3 = 1
x = -2
Ответ: x = -2.
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Answers & Comments
Verified answer
ОДЗ:25ˣ⁺³ - 1> 0
5ˣ⁺³ + 1> 0
25ˣ⁺³ > 25⁰
5ˣ⁺³ > -1 ⇒ x ∈ R
x + 3 > 0
x > -3
log₂(25ˣ⁺³ - 1) = 2 + log₂(5ˣ⁺³ + 1)
log₂(25ˣ⁺³ - 1) = log₂4 + log₂(5ˣ⁺³ + 1)
log₂(25ˣ⁺³ - 1) = log₂(4·5ˣ⁺³ + 4)
25ˣ⁺³ - 1 = 4·5ˣ⁺³ + 4
25ˣ⁺³ - 4·5ˣ⁺³ - 4 - 1 = 0
25ˣ⁺³ - 4·5ˣ⁺³ - 5 = 0
Пусть t = 5ˣ⁺³, t > 0
t² - 4t - 5 = 0
t² - 4t + 4 - 9 = 0
(t - 2)² - 3² = 0
(t - 2 - 3)(t - 2 + 3) = 0
(t - 5)(t + 1) = 0
t = 5; t = -1 - не подходит
Обратная замена:
5ˣ⁺³ = 5
5ˣ⁺³ = 5¹
x + 3 = 1
x = -2
Ответ: x = -2.