EXAMPLE OF SOLVING LOGARITHMIC INEQUALITIES

Solve the inequality.

Problem 1 :

Solution :

Solution is (125, ∞).

Problem 2 :

Solution :

The solution is (0, 64]

Problem 3 :

Solution :

Solution is [4, ∞).

Problem 4 :

Solution :

Solution is [4, ∞).

Problem 5 :

Solution :

Solution is [64, ∞).

Problem 6 :

Solution :

Solution is [81, ∞).

Problem 7 :

Solution :

Solution is (0, 108)

Problem 8 :

• Considering x > -7/3, all negative values should be ignored. The possible solution is (0, ∞).
• Considering x > -3/10, all negative values should be ignored. The possible solution is (0, ∞).
• Considering x > 4/7, the possible solution is (4/7, ∞).

By considering all three, the solution for the given inequality should be (4/7, ∞).

Problem 9 :

log x + log (2 - x) < 1

Solution :

log x + log(2 - x) < 1

log [x(2 - x)] < 1

Moving base to the other side of the inequality sign, we get

x(2 -x) < 10¹

2x - x² < 10

-x² + 2x - 10 < 0

x² - 2x + 10 > 0

This quadratic inequality cannot be solved using the method of factoring.

a = 1, b = -2 and c = 10

b² - 4ac = (-2)² - 4(1)(10)

= 4 - 40

= -36 < 0

Domain is (0, 2). So, the solution is (0, 2).

Problem 10 :

Solution :

Domain of log₂ (x² - x - 6)

x² - x - 6 > 0

(x - 3) (x + 2) > 0

x > 3 and x > -2

Domain of the function is (-2, ∞) u (3, ∞), solution by solving the logarithmic inequality is (-∞, 7).

By considering all these three, we get the solution as (3, 7).