Free Logarithm Calculator Online
This free online logarithm calculator handles three types of logarithms: common log (base 10), natural log (base e), and custom base. Every calculation shows all three results and includes inverse verification so you can confirm accuracy.
Must be a positive number
log₁₀(100) = ?
log₁₀ Result
Common logarithm of 100
Inverse Verification
10^2.000000 = 100
Logarithm Properties & Rules
The Core Definition
logₕ(x) = y means bⁿ = x
A logarithm is the inverse of exponentiation. The logarithm calculator answers: "To what power must the base b be raised to produce x?" For example, log₁₀(1000) = 3 because 10³ = 1000.
| Property | Formula | Example |
|---|---|---|
| Product Rule | log(ab) = log(a) + log(b) | log(2 × 5) = log(2) + log(5) = 1 |
| Quotient Rule | log(a/b) = log(a) − log(b) | log(100/10) = 2 − 1 = 1 |
| Power Rule | log(aⁿ) = n · log(a) | log(10³) = 3 · log(10) = 3 |
| Change of Base | logₕ(x) = ln(x) / ln(b) | log₂(8) = ln(8) / ln(2) = 3 |
| Log of 1 | logₕ(1) = 0 | ln(1) = 0, log(1) = 0 |
| Log of Base | logₕ(b) = 1 | log₁₀(10) = 1, ln(e) = 1 |
Worked Examples
Related Math Tools
Why Use This Free Logarithm Calculator?
- Three log modes (log, ln, custom)
- Inverse verification built in
- All results shown together
- Edge case handling
- Always free, no ads
Where Logarithms Show Up in Real Life
Sound & Decibels
The decibel scale is logarithmic. A sound at 80 dB is 10 times more intense than 70 dB, not just 10 units more. A jet engine at 140 dB is 10₁₄ (100 trillion) times more intense than the threshold of hearing. The logarithm calculator helps you work with these scales directly.
Earthquakes
The Richter scale uses log base 10. A magnitude 7 earthquake is 10 times stronger than a magnitude 6, and 1,000 times stronger than a magnitude 4. Each whole number increase represents a tenfold increase in measured amplitude.
Chemistry & pH
pH = -log₁₀([H⁺]). A solution with [H⁺] = 10⁻³ has pH 3 (acidic). Each pH unit represents a 10x change in hydrogen ion concentration. Use the logarithm calculator to convert between concentration and pH.
Common Logarithm Mistakes
These errors come up frequently in math classes, standardized tests, and practical applications.
log(a + b) is NOT log(a) + log(b)
The product rule says log(a × b) = log(a) + log(b). There is no equivalent rule for addition. log(3 + 7) = log(10) = 1, but log(3) + log(7) = 0.477 + 0.845 = 1.322.
Confusing log and ln Bases
In pure math and many European countries, "log" often means natural log (ln). In engineering and U.S. textbooks, "log" usually means base 10. Always check which convention is being used. The logarithm calculator shows both to avoid ambiguity.
Attempting log of Zero or Negative Numbers
log(0) is undefined (it approaches negative infinity), and log of a negative number has no real solution. The logarithm calculator validates input and warns you if x is not positive.
Frequently Asked Questions
Disclaimer: This logarithm calculator is for educational and informational purposes only. Results are computed using JavaScript floating-point math and may have minor rounding differences from exact values.
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