Logarithmic Inequalities

Logarithmic inequalities involve expressions containing logarithms and require careful attention to:

• domain restrictions

• properties of logarithmic functions

• monotonic behavior of logarithms

Because logarithmic functions are strictly increasing or decreasing (depending on the base), they allow inequalities to be solved systematically, but errors in domain handling are common.

Behavior of Logarithmic Functions

For $$f(x)=\log_{a}x$$:

• Domain: x>0

• If a>1:
  $$\log_{a}x$$ is increasing

• If 0<a<1:
  $$\log_{a}x$$ is decreasing

This behavior determines whether an inequality sign is preserved or reversed.

Domain Restrictions (Critical First Step)

Before solving any logarithmic inequality:

The argument of every logarithm must be positive.

Examples:

• $$\log(x-2)>1 \Rightarrow x > 2$$

• $$\ln(3x+1)\le2 \Rightarrow x > -\frac{1}{3}$$​

Domain restrictions must be applied before and after solving.

Solving Simple Logarithmic Inequalities

Single Logarithm, Base Greater Than 1

Example:

$$\log_2(x-1)>3$$

Step 1: Domain

$$x-1>0 \rightarrow x>1$$

Step 2: Rewrite exponentially

$$x-1>2^3 \rightarrow x>9$$

Final solution:

x>9

Base Between 0 and 1 (Inequality Reversal)

Example:

$$\log_{\frac{1}{2}}(x+4)<1$$

Because the base is between 0 and 1, the function is decreasing, so the inequality reverses:

$$x+4>\left(\frac{1}{2}\right)^1 \rightarrow x>-\frac{7}{2}$$​

Also apply domain:

$$x>-4$$

Final solution:

$$x>-\frac{7}{2}$$​

Logarithmic Inequalities with Multiple Log Terms

Example:

$$\log(x+2)-\log(x-1)\ge1$$

Step 1: Domain

$$x+2>0 , x-1>0 \rightarrow x>1$$

Step 2: Combine using log rules

$$\log\left(\frac{x+2}{x-1}\right)\ge1$$

Step 3: Rewrite exponentially

$$\frac{x+2}{x-1}\ge10^1$$

Step 4: Solve rational inequality

$$x+2\ge10(x-1) \rightarrow x\le\frac{12}{9}=\frac{4}{3}$$​

Step 5: Apply domain

$$1<x\le\frac{4}{3}$$​

Logarithmic Inequalities with Different Bases

When bases differ, use change of base to rewrite:

$$\frac{\ln(x-1)}{\ln2}>3$$

Since ln⁡2>0, the inequality direction remains unchanged.

Common Errors

• Ignoring domain restrictions

• Forgetting to reverse inequality when 0<a<1

• Including extraneous solutions

• Misapplying logarithm laws

• Solving logarithmic inequalities like equations

AP Exam Focus

Students should be able to:

• justify inequality direction using monotonicity

• apply logarithm laws correctly

• solve and interpret solutions using interval notation

• explain reasoning verbally or in writing

• check solutions against domain restrictions

Summary

Logarithmic inequalities require:

1. Careful domain analysis

2. Understanding of logarithm behavior

3. Correct use of exponential equivalence

4. Verification of solutions