Log Calculator
Calculate the logarithm of any number in any base – common log (base 10), natural log (ln), binary log (base 2) or your own – and the antilog.
What a logarithm is
The logarithm answers the question “to which power must I raise the base to get this number?”. log₁₀ 1,000 = 3 because 10³ = 1,000. log₂ 8 = 3 because 2³ = 8. In general, log_b(x) = y means by = x.
The common bases
| Name | Base | Example |
|---|---|---|
| Common logarithm (log) | 10 | log 100 = 2 |
| Natural logarithm (ln) | e ≈ 2.718282 | ln 10 ≈ 2.302585 |
| Binary logarithm | 2 | log₂ 1,024 = 10 |
Change of base
Any logarithm can be calculated with natural logs: log_b(x) = ln(x) / ln(b). For log₅ 125: ln 125 ≈ 4.828314 and ln 5 ≈ 1.609438, and 4.828314 / 1.609438 = 3. That is how the calculator works for any base, and why it shows the steps.
Rules of logarithms
- log(a · b) = log a + log b
- log(a / b) = log a − log b
- log(aⁿ) = n · log a
- log_b(b) = 1 and log_b(1) = 0
Antilog
The antilogarithm goes the other way: if log x = 3, then x = 10³ = 1,000. Choose Number (antilog) to calculate by. Logarithms are only defined for positive numbers, and the base must be positive and not 1.
Frequently asked questions
What is log base 10 of 1000?
3, because 10³ = 1,000.
What is the difference between log and ln?
log usually means base 10, ln means base e ≈ 2.718282. ln x = log x × 2.302585.
Can you take the log of a negative number?
Not with real numbers. There is no real power of a positive base that gives 0 or a negative number.