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