# Logarithmic Inequalities

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

## Example 1: Solve log_{7} (*x* + 2) ≤ 1

Before solving the inequality,

write the condition of the domain.

The domain of log_{7} (*x* + 2) is

(*x* + 2).

So *x* + 2 > 0.

So *x* > -2.

Logarithmic equations

Solve the given inequality.

Solve the log.

(base: 7)

Then *x* + 2 ≤ 7^{1}.

The base [7] satisfies

[base] > 1.

So the order of the inequality sign

does not change.

Logarithmic form

7^{1} = 7

Move +2 to the right side.

Then *x* ≤ 5.

*x* > -2*x* ≤ 5

Draw these two inequalities

on a number line.

Then the intersecting region is

-2 < *x* ≤ 5.

## Example 2: Solve log_{0.1} (*x* - 3) > 2

Before solving the inequality,

write the condition of the domain.

The domain of log_{0.1} (*x* - 3) is

(*x* - 3).

So *x* - 3 > 0.

So *x* > 3.

Solve the given inequality.

Solve the log.

(base: 0.1)

Then *x* - 3 < 0.1^{2}.

The base [0.1] satisfies

0 < [base] < 1.

So the order of the inequality sign

does change.

0.1^{2} = 0.01

Move -3 to the right side.

Then *x* < 3.01.

*x* > 3*x* < 3.01

Draw these two inequalities

on a number line.

Then the intersecting region is

3 < *x* < 3.01.

## Example 3: Solve 2 log_{3} *x* ≥ log_{3} (*x* + 6) + 1

Before solving the inequality,

write the conditions of the domains.

First, the domain of log_{3} *x* is*x*.

So *x* > 0.

Next, the domain of log_{3} (*x* + 6) is

(*x* + 6).

So *x* + 6 > 0.

So *x* > -6.

*x* > 0*x* > -6

Find the intersection of these two inequalities

by drawing the inequalities

on a number line.

Then *x* > 0.

Solve the given inequality.

2 log_{3} *x* = log_{3} *x*^{2}

Logarithm of a power

+1 = +log_{3} 3

Logarithm of the base

log_{3} (*x* + 6) + log_{3} 3 = log_{3} (*x* + 6)⋅3

Logarithm of a product

The bases of the logs on both sides

are the same: 3.

Then *x*^{2} ≥ (*x* + 6)⋅3.

The base [3] satisfies

[base] > 1.

So the order of the inequality sign

does not change.

(*x* + 6)⋅3 = 3*x* + 18

Move 3*x* + 18 to the left side.

Then *x*^{2} - 3*x* - 18 ≥ 0.

Factor *x*^{2} - 3*x* - 18.

Factor a quadratic trinomial

Find a pair of numbers

whose product is the constant term [-18]

and whose sum is the middle term's coefficient [-3].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

(-18, 1) and (-9, 2)

are not the right numbers.

[-18] = -6⋅3

-6 + 3 = [-3]

So -6 and 3 are the right numbers.

Use -6 and +3

to write the factored form:

(*x* - 6)(*x* + 3) ≥ 0.

Find the zeros of (*x* - 6)(*x* + 3) ≥ 0.

Solving a quadratic equation by factoring

1) *x* - 6 = 0

So *x* = 6.

2) *x* + 3 = 0

So *x* = -3.

So the zeros are *x* = 6, -3.

To find the range of the inequality,

draw *y* = (*x* - 6)(*x* + 3)

on the *x*-axis.

The coefficient of *x*^{2} is (+): 1.

And its zeros are -3 and 6.

So draw a parabola

that is opened upward

and that passes through *x* = -3, 6 on the *x*-axis.

Solving quadratic inequalities

Find the range that satisfies*y* = (*x* - 6)(*x* + 3) ≥ 0.*y* is [greater than or equal to] 0.

So color the [upper region] of the graph,

[including the zeros].

Then *x* ≤ -3 or *x* ≥ 6.

*x* > 0

(from the domains condition)*x* ≤ -3 or *x* ≥ 6

(the solution of the inequality)

Draw these two inequalities

on a number line.

Then the intersecting region is*x* ≥ 6.