Enter the inequality
Interactive graph verification
Formula used
How to use this calculator
- Enter the complete left and right expressions.
- Select the required inequality sign.
- Use explicit multiplication, such as
2*x. - Choose a search range containing all expected boundaries.
- Select symbolic, numerical, or automatic solving.
- Press Solve inequality and review the domain first.
- Use the graph, sign chart, and export tools for verification.
Supported syntax
| Purpose | Accepted input | Example |
|---|---|---|
| Custom-base logarithm | log(value, base) | log(x-1, 3) |
| Natural logarithm | ln(value) | ln(x+4) |
| Base-ten logarithm | log(value), lg(value), or log10(value) | log10(2*x-5) |
| Named base | log2(value) or log3(value) | log2(x^2-1) |
| Powers | ^ or pow(a,b) | x^2 |
| Other functions | abs, sqrt, exp, sin, cos | ln(exp(x)-1) |
Example data
| Inequality | Domain idea | Expected solution |
|---|---|---|
log2(x-1) > 3 | x > 1 | x > 9 |
ln(x+4) ≤ 2 | x > -4 | -4 < x ≤ e²-4 |
log(x+2,0.5) > -1 | x > -2 | -2 < x < 0 |
log2(x)+log2(x-2) ≥ 3 | x > 2 | x ≥ 4 |
log10(x²-1) ≥ 1 | |x| > 1 | x ≤ -√11 or x ≥ √11 |
Understanding logarithmic inequalities
Logarithmic inequalities compare expressions containing one or more logarithms. Their solutions depend on algebra and strict domain restrictions. Every logarithm argument must remain positive throughout the solution.
A logarithm base must also satisfy two conditions. It must be positive, and it cannot equal one. These rules prevent undefined or meaningless logarithmic expressions.
Bases greater than one create increasing logarithmic functions. Increasing functions preserve the original inequality direction. This property often reduces logarithmic comparisons to algebraic comparisons.
Bases between zero and one behave differently. Their logarithmic functions decrease as inputs increase. Therefore, removing equal logarithms reverses the inequality sign.
Product rules can combine sums of compatible logarithms. Quotient rules can combine logarithmic differences when domains remain valid. Power rules move coefficients into logarithm arguments as exponents.
Domain restrictions must be found before any transformation. Squaring or multiplying can introduce invalid candidate values. Final answers must always be intersected with the original domain.
Complex expressions may produce several critical boundaries. These can include zeros, asymptotes, and logarithm domain endpoints. A sign chart tests one point within each resulting interval.
Strict inequalities use open endpoints at equality boundaries. Inclusive inequalities may use closed endpoints when expressions remain defined. Domain boundaries always stay open because logarithms exclude zero arguments.
Numerical solving helps when symbolic isolation becomes difficult. A careful scan finds truth changes across the selected range. Bisection then refines each detected transition to higher precision.
Graphs provide useful confirmation but not formal proof. Narrow intervals or tangent roots may be visually subtle. The sign chart and original-domain checks remain essential.
This calculator combines symbolic recognition with numerical interval testing. Simple linear logarithmic forms receive an exact boundary statement. More complicated forms use controlled real-number sampling and refinement.
Choose a wide enough search range for every expected solution. Narrow the range when close boundaries require better resolution. Careful domain checks always prevent misleading logarithmic inequality answers.
Frequently asked questions
Why must logarithm arguments be positive?
Real logarithms are defined only for positive inputs. Zero and negative arguments are excluded from every solution set.
When does the inequality sign reverse?
It reverses when applying a decreasing logarithmic function with a base strictly between zero and one.
Can the calculator solve logarithms on both sides?
Yes. Enter complete logarithmic expressions on both sides, then choose the desired comparison sign.
Why is the answer limited to the selected range?
The general solver searches numerically. Increase the range when solutions may exist outside the displayed interval.
Does the calculator always return exact answers?
It returns exact boundary forms for recognized simple cases. General expressions receive high-precision numerical boundaries.
Can I enter a base below one?
Yes. Use syntax such as log(x+2,0.5). The solver accounts for decreasing behavior.
What causes an undefined interval?
Common causes include nonpositive logarithm arguments, invalid bases, division by zero, or negative square-root inputs.
Can I restrict answers to integers?
Yes. Select Integers only, and the calculator lists valid integer values inside your chosen range.
How can I improve numerical accuracy?
Use a narrower search range, reduce the numerical step, and increase the displayed decimal precision.