Question

Решите логарифмические уравнения

Answer 1

1.

{x²-2x-9 = x+9

{x+9 > 0; x > -9

x²-3x-18 = 0

{x1+x2 = 3

{x1•x2 = -18

x1 = 6

x2 = -3

Ответ: {-3; 6}

2.

log7(x-4) + log7(x+1) = log7(4(x+1))

log7(x-4) + log7(x+1) = log7(4) + log7(x+1)

{log7(x-4) = log7(4)

{x+1 > 0; x > -1

{x-4 = 4

{x > -1

x = 8

Ответ: {8}

Answer 2

1)

log5(x^2 - 2x -9) = log5(x+9)

ОДЗ:

1) (x^2 - 2x -9)>0  

2) (x+9)>0

x^2 - 2x - 9 = x + 9

x^2 - 3x - 18 = 0

D = 81 = 9^2

x1 = (3+9)/2 = 6

x2 = (3-9)/2 = -3

Ответ: 6 или -3

2)

log7(x-4) + log7(x+1) = log7(4x+4)

ОДЗ:

1) x-4 >0

2) x+1 >0

3) 4x+4 >0

log7( (x-4)(x+1) ) = log7(4x+4)

x^2 - 4x + x - 4 = 4x + 4

x^2 - 7x - 8 = 0

D = 49 + 32 = 81 = 9^2

x1 = (7+9)/2 = 8

x2 = (7-9)/2 = -1  - Не соответствует ОДЗ.

Ответ: 8