# Logarithmic Inequalities

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

## Example 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

(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

(gray) should be (+).

So *x* > 0 and *x* + 6 > 0.

The intersection is *x* > 0.

Left side

Logarithms of powers

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.