The short answer
A logarithm answers "to what power must the base be raised to get this number?" log₂(8) = 3 because 2³ = 8. Antilog reverses it: given the log value and base, it returns the original number. This calculator handles natural log (ln, base e), common log (base 10), and any custom base, in both directions.
Key takeaways
- log_b(x) and bʸ are inverse operations — plugging a log result back into the exponential form should return the original number.
- The domain of a logarithm is positive numbers only; log of zero or a negative number has no real answer.
- Any logarithm can be computed in any other base using the change-of-base formula: log_b(x) = log(x) ÷ log(b).
- Log rules turn multiplication into addition and exponents into multiplication, which is why logarithms are central to measuring scales like pH and decibels.
The three log rules that matter most
| Rule | Formula | What it means |
|---|---|---|
| Product rule | log(ab) = log(a) + log(b) | Multiplication becomes addition |
| Quotient rule | log(a/b) = log(a) − log(b) | Division becomes subtraction |
| Power rule | log(aⁿ) = n·log(a) | Exponents become multiplication |
Change of base formula
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Most calculators only have built-in buttons for natural log and log base 10 — the change-of-base formula lets you compute a logarithm in any other base by dividing two logs you can already calculate directly.
Worked example: solving with change of base
log₂(8) = ln(8) / ln(2)
= 2.0794 / 0.6931
= 3
This matches the direct answer, since 2³ = 8 — the change-of-base formula gives the same result as computing log₂(8) directly, but it works even when the base isn't one you can solve by inspection.
Common mistakes to avoid
- Trying to take the log of zero or a negative number — logarithms are only defined for positive real arguments.
- Confusing log (base 10) with ln (base e) — they give different results for the same input: log(100) = 2, but ln(100) ≈ 4.6052.
- Treating log(a + b) as log(a) + log(b) — the product rule only applies to multiplication; there's no simple rule for the log of a sum.
- Using an antilog base that doesn't match the log's original base — that mismatch won't correctly reverse the calculation.
Related calculators
- Exponent Calculator — work the inverse direction: powers instead of logs.
- Root Calculator — solve for roots, another operation closely tied to exponents.
- Half-Life Calculator — see natural log at work in exponential decay problems.
- Scientific Notation Calculator — connect log base 10 to orders of magnitude.