Solving logarithmic inequalities

logarithmic inequality is an inequality in which the unknown value is under the sign of the logarithm.

Formulas for logarithmic inequalities

1. The value of the logarithm is greater than zero (log_a x > 0) provided that both the base and the sublogarithmic expression are on the same side of the number 1. There can be two options here:

• a> 1 и x> 1
• 0 и 0

Accordingly, if a и x stand on opposite sides of unity, the value of the logarithm log_ax negative.

2. For the logarithmic inequality log_a f(x) > b rightly:

• f(x) < a^b with 0
• f(x) > a^b with a> 1

Similarly, for log_af(x) < b right:

• f(x) > a^b with 0
• f(x) < a^b with a> 1

3. Inequality of the species log_a f(x) > log_a g (x) comes down to:

• 0 < f(x) < g(x) with 0
• f(x) > g(x) > 0 with a> 1

Likewise, for log_a f(x) < log_a g (x) it can be argued:

• f (x) > g(x) > 0 with 0
• 0 < f(x) < g(x) with a> 1

Task Examples

Task 1

Solve the inequality log_0,7(x-3) > 3.

Decision:

The base of the logarithm is greater than zero but less than one (0<0,7<1). Applying the corresponding formula (f(x) < a^b with 0), we get:

(x-3) < 0,7³

(x-3) < 0,343
x <3,343

At the same time, the sublogarithmic expression of any logarithm must be greater than zero. Therefore, (x-3) > 0, which means x>3.

Thus, by combining both conditions, we determine x∈(3;3,343).

Task 2

Solve the inequality log₂82x.

Decision:

Because the base of the logarithm is greater than one, for the given inequality is true: 0<8