Check the logarithm domain first
The argument of a logarithm must be positive before any logarithm rule is applied.
This condition is required. Values outside it are illegal even if they satisfy the later algebra.
Logarithmic Inequality Calculator
Solve logarithmic inequalities step-by-step — including the domain check most calculators skip.
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Why this page is different
The argument domain f(x) > 0 is shown before the algebra, not hidden in a later note.
The base test separates a > 1 from 0 < a < 1, so fractional bases visibly trigger the sign flip.
The safety checker compares the common wrong answer with the domain-safe final interval.
Interactive Calculator
Enter the logarithmic inequality
The domain check is locked as Step 1, then the raw inequality answer is intersected with the valid log domain.
One-tap examples
Logarithm compared with a constant
Domain check first: x-3 > 0. This step is never skipped.
Common mistake
Stop after solving the transformed inequality and report:
Correct method
Intersect the raw answer with the log domain:
Step-by-Step Breakdown
Read the base direction
Because the base is greater than 1, the logarithm is increasing and the inequality direction stays the same.
Rewrite the constant as a same-base logarithm
The constant 2 is the same as \log_10(100).
Remove the logarithm and solve
After matching the base, compare the arguments using the base direction from the previous step.
Intersect with the domain
The final answer keeps only values that satisfy both the algebraic solution and the domain restriction.
Teacher note: Many students stop at (-∞, 103). The correct answer is (3, 103) after the domain check.
Shareable setup
https://inequalitycalculator.net/logarithmic-inequality-calculator?mode=log-constant&base=10&left=x+-+3&relation=%3C&constant=2Calculator Types
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Method
How the domain-safe solver works
Choose whether your problem compares one logarithm with a constant, two logarithms with the same base, or two logarithms with different bases.
Enter the base, logarithm argument, inequality sign, and right-side value or logarithm.
Read the domain check first, then compare the raw algebraic solution with the domain-safe final answer.
Many students solve the transformed inequality correctly, then lose points because they forget that every logarithm argument must stay positive in the original problem.
Problem Types
Match the log structure before comparing anything
log_a(f(x)) > c
One log compared with a constant
Require f(x) > 0 before solving.
Rewrite the constant as a power, then compare f(x) with a^c. Flip the comparison only when 0 < a < 1.
log_a(f(x)) <= log_a(g(x))
Two logs with the same base
Require both f(x) > 0 and g(x) > 0.
Compare the arguments directly, using the base direction to decide whether the inequality sign stays or flips.
log_a(f(x)) > log_b(g(x))
Two logs with different bases
Intersect all positive-argument conditions.
Use change of base, solve the transformed inequality, then keep only values that still satisfy the original domains.
Common Mistakes
Domain and base mistakes to catch early
Solving the transformed inequality first
Write every log argument domain first, then intersect the final algebraic interval with those conditions.
Treating base 1/2 like base 2
A base between 0 and 1 makes the logarithm decreasing, so argument comparisons reverse.
Canceling logs with different bases
Only same-base logarithms can be compared by arguments. Different bases need change-of-base work.
Examples
Logarithmic inequality examples with domain checks
Base greater than 1
log_2(x - 1) > 3
Domain x > 1 is checked first, then x - 1 > 8 gives x > 9. The domain is already contained in the stricter result.
Domain
Raw solution
Final answer
Fractional base sign flip
log_(1/2)(x + 2) < -2
Base 1/2 is between 0 and 1, so the sign flips. Domain x > -2 and algebra x > 2 intersect to x > 2.
Domain
Raw solution
Final answer
Different bases
log_2(x) > log_4(x + 3)
Change of base rewrites the comparison, then the solution is intersected with x > 0 and x + 3 > 0.
Domain
Raw solution
Final answer
FAQ
Frequently Asked Questions
What are the conditions for solving logarithmic inequalities?
Every logarithm must have a valid base and a positive argument. The base must be greater than 0 and cannot equal 1. Each argument must be greater than 0, and the final solution must be intersected with those domain conditions.
Why do you need to check the domain when solving log inequalities?
A logarithm is undefined when its argument is zero or negative. Solving the transformed inequality can produce values that look correct algebraically but are illegal in the original logarithmic expression, so the domain check must happen before the final answer.
When do you flip the inequality sign in logarithmic inequalities?
If the base is greater than 1, the logarithm is increasing and the inequality direction stays the same. If the base is between 0 and 1, the logarithm is decreasing and the inequality direction flips when you compare the arguments.
Can the argument of a logarithm be negative or zero?
No. The argument of a logarithm must be positive. For example, log(x - 3) requires x - 3 > 0, so x must be greater than 3 before any other solution step is accepted.
How do you solve a logarithmic inequality with different bases?
Use the change-of-base formula to rewrite both logarithms with a common logarithm or natural logarithm, then solve the resulting comparison on the valid domain. The calculator shows this step explicitly for supported linear-argument patterns.
How is solving a log inequality different from a log equation?
Both require domain checks, but inequalities also need direction logic. A base between 0 and 1 reverses the inequality, and the answer is usually an interval rather than one or two isolated values.