Logarithm Calculator
log, ln, log₂, and any custom base, with change of base shown.
Means: 10^3 = 1000
A logarithm asks what power a base must be raised to: since 10³ = 1000, log₁₀(1000) = 3. For any base, use change of base: logᵦ(x) = ln(x) ÷ ln(b). So log₂(8) = ln 8 ÷ ln 2 = 3, because 2³ = 8.
The inverse of exponentiation
A logarithm answers a single question: “to what power must I raise the base to get this number?” If 10³ = 1000, then log₁₀(1000) = 3. Logarithms turn multiplication into addition, which is why they underpin slide rules, decibels, the Richter scale, and pH. To compute a log in any base, the change-of-base formula reduces it to logs your calculator already knows.
change of base; ln can be swapped for log₁₀ — any common base works
Worked example
Compute log₂(8) using change of base:
- 1 Rewrite with natural logs. log₂(8) = ln(8) ÷ ln(2).
- 2 Evaluate each log. ln(8) ≈ 2.0794 and ln(2) ≈ 0.6931.
- 3 Divide. 2.0794 ÷ 0.6931 = 3. Check: 2³ = 8. ✓
Logarithm identities
The core rules — they turn products into sums and powers into multipliers.
| Identity | Rule | Example |
|---|---|---|
| Product | log(xy) = log x + log y | log(100) = log(10) + log(10) = 2 |
| Quotient | log(x ÷ y) = log x − log y | log₂(8 ÷ 2) = 3 − 1 = 2 |
| Power | log(xⁿ) = n · log x | log(10³) = 3 · log(10) = 3 |
| Change of base | log_b(x) = ln x ÷ ln b | log₂(8) = 2.0794 ÷ 0.6931 = 3 |
| Base of itself | log_b(b) = 1 | log₅(5) = 1 |
Domain and the three named bases
The input must be positive. No power of a positive base produces zero or a negative number, so logarithms of x ≤ 0 are undefined over the reals.
The base can’t be 1 (or ≤ 0). 1 raised to any power is always 1, so log base 1 could never reach other values.
log, ln, log₂. “log” usually means base 10; “ln” is the natural log, base e ≈ 2.718; “log₂” is base 2, common in computing. The change-of-base formula links them all.