Log calculator
Logarithms in base 10, e, 2 or any base, and powers (antilog), with change of base steps.
- ๐บ๐ธ US federal & state rules
- See sources
log(1000)
3
- log base 10
- 3
- Natural log (ln)
- 6.90775527898
- log base 2
- 9.96578428466
This is an example. Enter your numbers above and press the button to see your own result.
Size in powers of 10
| Characteristic (integer part of log) | 3 |
|---|---|
| Mantissa | 0 |
| Digits before the decimal point | 4 |
| Scientific notation | 1 ร 10^3 |
Show the math
| Definition | log(1000) = y means 10^y = 1000 |
|---|---|
| Change of base | log(1000) = ln 1000 รท ln 10 |
| Natural logs | ln 1000 โ 6.90775527898, ln 10 โ 2.30258509299 |
| Divide | โ 3 |
| Exact check | 10^(3) = 1000 |
What to do next
How it works
log_b x is the exponent y with b^y = x. The calculator uses ln x รท ln b and then checks whether the answer is a simple fraction, so log base 8 of 2 shows as 1/3.
For a power b^y with whole numbers, the exact value is computed digit by digit, so 2^100 shows all 31 digits. Very large or tiny results are shown in scientific notation.
Things to keep in mind
The base must be positive and not 1, and the number must be positive. Results are rounded to 12 significant digits.
Sources
- NIST DLMF 4.2, Logarithm: definitionsChecked 2026-10-19
Frequently asked questions
What is the difference between log and ln?
log usually means base 10 and ln means base e (about 2.71828). log 100 = 2 because 10ยฒ = 100, and ln e = 1.
How do I calculate a log with a different base?
Use change of base: log_b x = ln x รท ln b. log base 3 of 81 is ln 81 รท ln 3 = 4.
What is an antilog?
It undoes a log. The antilog of 3 in base 10 is 10ยณ = 1,000. Pick Power to compute b^y.
Why can't I take the log of a negative number?
No real power of a positive base gives a negative number or 0, so the log is undefined in the real numbers.