2 * 4^x+1 + 15 * 2^x - 2 = 0
2 * 4^x * 4 + 15 * 2^x - 2 = 0
Замена:
2^x = a
8a^2 + 15a - 2 = 0
D = 225 + 64 = 289
D = 17
-15+--17/2
x1 = 2; x2 = -16
Второй корень не удовлетворяет. Возвращаемся к замене.
2^x = 2^1
x = 1
3. y= sqrt(5^3x+1 -1)
5^3x + 1 >= 1
5^3x + 1 >= 5^0
3x + 1 >= 0
3x >= - 1
x >= -1/3
(-1/3 ; + бесконечности)
4. log^2_3 (x) - 2log_3 (x) = 3
log_3 (x) = a
a^2 - 2a - 3 = 0
D = 4 + 12= 16
D = 4
2 +-4/2
x1 = 3; x2 = -1
Возвращаемся к замене.
log_3 (x) = 3
x = 3^3
x = 27
log_3 (x) = -1
x = 3^-1
x = 1/3
{1/3;27}
4.1 log_5 (2x +3) + log_5 (4-x) = log_5 (5)
log5(2x+3)(4-x) = log5(5)
ОДЗ:
2x +3 > 0
4 - x > 0
x > -3/2
x < 4
2x^2 - 5x - 7 = 0
D = 25 + 56
D = 9
5 +-9/4
x1 = 14/4
x2 = -1
Оба корня удовлетворяют ОДЗ
{-1;14/4}
5.log5(x) < 2
x < 25
5.1 log0,2 (x/7) = 0
x/7 > 1
x > 7
log1/4 x > log1/4 5x- 4
4x > 4
x > 1