A) log₀,₅ (x²-3x) = - 2 b) log₂² (x-2) - log₂ (x-2) = 2 c) log₃ (x²+2x) <1 d) log₁/₃ (0,1x-5,2) >2

A) Найдем корни уравнения.

log₀,₅ (x² - 3 * x) = - 2;

ОДЗ:

x² - 3 * x > 0;

x * (x - 3) > 0;

Отсюда, x 3.

b) log₂² (x - 2) - log₂ (x - 2) = 2;

log₂² (x - 2) - log₂ (x - 2) - 2 = 0;

x > 2;

log₂ (x - 2) = а;

a² - а - 2 = 0;

D = 1 - 4 * 1 * (-2) = 3²;

a1 = (1 + 3) / 2 = 2;

a2 = (1 - 3) / 2 = - 1;

{ log₂ (x - 2) = 2;

log₂ (x - 2) = - 1;

{ x - 2 = 4;

x - 2 = 1/2;

{ x = 6;

x = 3.5.

c) log₃ (x² + 2 * x) < 1;

{ x² + 2 * x > 0;

x² + 2 * x < 3;

{ x 0;

x² + 2 * x - 3 < 0;

{ x 0;

-3 < x < 1;

Отсюда, - 3 < x < - 2 и 0 < x < 1.

d) log₁/₃ (0,1 * x - 5,2) > 2;

{ 0.1 * x - 5.2 > 0;

0.1 * x - 5.2 < 1/9;

{ x > 52;

x - 52 > 10/9;

{ x > 52;

x > 52 + 1 1/9;

{ x > 52;

x > 53 1/9;

Отсюда, x > 53 1/9.