Logarithmic Inequalities

Logarithmic Inequalities

How to solve logarithmic inequalities: examples, and their solutions.

Example 1

Solve the given inequality. log_7 (x + 2) <= 1

(gray) should be (+).
So x + 2 > 0.

x > -2

Solve the log.

Logarithmic form

The base (7) is greater than 1.
So the order of the inequality sign
doesn't change.

Find the intersecting region of
x > -2 and x ≤ 5.

-2 < x ≤ 5

Example 2

Solve the given inequality. log_0.1 (x - 3) > 2

(gray) should be (+).
So x - 3 > 0.

x > 3

Solve the log.

Logarithmic form

The base (0.1) is between 0 than 1.
So the order of the inequality sign
does change.

Find the intersecting region of
x > 3 and x < 3.01.

3 < x < 3.01

Example 3

Solve the given inequality. 2 log_3 x >= log_3 (x + 6) + 1

(gray) should be (+).
So x > 0 and x + 6 > 0.

The intersection is x > 0.

Left side
Logarithms of powers

Logarithms of products

Solve the log.

Logarithmic form

The base (3) is greater than 1.
So the order of the inequality sign
doesn't change.

Solve the quadratic inequality.

x < -3 or x > 6

Solving quadratic inequalities

Find the intersecting region of
x > 0 and 'x < -3 or x > 6'.

x > 6 is the answer.