Free log calculator
Calculate logarithms in any base — enter a value and base to see log₁₀, ln, or any custom base result, updated live, as you type.
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log_b(x) = ln(x) / ln(b). x must be > 0; base must be > 0 and ≠ 1.
Results are estimates. Consult a professional.
How the logarithm calculator works
A logarithm answers the question: "to what power must I raise the base to get this number?" It is the inverse of exponentiation. Logarithms compress huge ranges (like earthquake magnitudes or sound levels) into manageable scales and appear throughout science, engineering, and finance.
Key identities: log(xy) = log(x) + log(y); log(x/y) = log(x) − log(y); log(xⁿ) = n·log(x). These turn multiplication into addition, which is why logarithms were the original calculating tool before electronic computers.
NIST Digital Library of Mathematical Functions — §4.1 Special Values and IdentitiesWorked example: log₁₀, ln, and log₂
Three logarithms covering the most common bases: base 10, base e, and base 2.
Common logarithm values
The table below covers the most-referenced log₁₀ and ln values. Memorising a few anchors (log 2 ≈ 0.301, log 3 ≈ 0.477) lets you estimate most others.
| x | log₁₀(x) | ln(x) | Note |
|---|---|---|---|
| 0.001 | −3 | −6.908 | 10⁻³ |
| 0.01 | −2 | −4.605 | 10⁻² |
| 0.1 | −1 | −2.303 | 10⁻¹ |
| 1 | 0 | 0 | log of 1 is always 0 |
| 2 | 0.301 | 0.693 | log 2 — key anchor |
| e | 0.434 | 1 | ln(e) = 1 by definition |
| 10 | 1 | 2.303 | 10¹ |
| 100 | 2 | 4.605 | 10² |
| 1,000 | 3 | 6.908 | 10³ |
| 10,000 | 4 | 9.210 | 10⁴ |
Values rounded to three decimal places. ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.3026.
Tips for working with logarithms
Logarithms are counterintuitive at first but follow consistent rules. These tips cover the most common stumbling blocks.
- The domain is positive reals only — log(0) is undefined (approaches −∞); log of a negative number is not a real number; always check your input is positive.
- Use change of base for any base — most calculators have only log₁₀ and ln; compute log_b(x) = ln(x)/ln(b) or log(x)/log(b) for any other base.
- Adding logs is multiplying arguments — log(a) + log(b) = log(a×b); this is why log tables and slide rules could turn multiplication into addition.
- Decibels and the Richter scale use log₁₀ — a 3 dB increase doubles power (10^0.3 ≈ 2); a magnitude 7 earthquake is 10× more intense than magnitude 6.
- ln appears in exponential growth/decay — if a quantity grows as e^(kt), then ln gives you the rate k; half-life, compound interest, and population models all use ln.
Accuracy and limitations
This calculator uses the JavaScript Math.log() function (natural logarithm) internally and applies the change-of-base formula for other bases. Results are accurate to approximately 15 significant figures across the full positive real range. Very small inputs close to zero (e.g., x = 10⁻³⁰⁰) and very large inputs (e.g., x = 10³⁰⁰) remain within the double-precision range. Inputs of zero or negative numbers are invalid and the calculator will display an error.
Key terms
About this calculator
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