Question

1ый пример: log2(3-x)=0 2ой пример : lg(3x²+12x+19)-lg(3x+4)=1 3ий пример : log11(x+4)+log11(x-7)=log11(7-x) 4ый пример : log2²x-4 log2x+3=0

Answer 1

1. log₂(3 - x) = log₂1 ОДЗ: 3 - x > 0 | x < 3
3 - x = 1
- x = - 2
x = 2
ответ: x = 2.
2. lg(3x² + 12x + 19) - lg(3x + 4) = lg10
$lg\frac{3x^2 + 12x + 19}{3x + 4} = lg10 \\ \frac{3x^2 + 12x + 19}{3x + 4} = 10$
3x² + 12x + 19 = 10(3x + 4)
ОДЗ: 3x - 4 ≠ 0 | x ≠ 4/3 
3x² + 12x + 19 = 30x + 40
 3x² - 18x - 21 = 0 | /3
x² - 6x - 7 = 0
D = 36 - 4*1*(-7) = 36 + 28 = 64 = 8²
x₁ = (6 + 8)/2 = 7 x₂ = (6 - 8)/2 = - 1
ОДЗ: 3x + 4 > 0
x > - 4/3
x∈(-4/3; + бесконечность)
ответ: x = - 1; 7.
3. log₁₁((x + 4)*(x - 7)) = log₁₁(7 - x)
(x + 4)(x - 7) = 7 - x
x² - 7x + 4x - 28 = 7 - x
x² - 3x + x - 28 - 7 = 0
x² - 2x - 35 = 0
D = 4 - 4*1*(-35) = 4 + 140 = 144 = 12²
x₁ = (2 + 12)/2 = 7 x₂ = (2 - 12)/2 = - 5
ОДЗ: x + 4 > 0 
          x - 7 > 0
          7 - x > 0
x∈(7; +бесконечность)
ответ: x∈∅
4. log₂x = t
t² - 4t + 3 = 0
D = 16 - 4*1*3 = 16 - 12 = 4 = 2²
x₁ = (4 + 2)/2 = 3 x₂ = (4 - 2)/2 = 1
$\left \{ {{log_{2}x = 3 } \atop {log_2x = 1}} \right. \\ \left \{ {{x = 2^3} \atop {x=2^1}} \right. \\ \left \{ {{x=8} \atop {x=2}} \right.$
ОДЗ: x > 0
ответ: x = 2; 8.