Logarithm Calculator
The logarithm of a number in base 10, base 2, the natural base e or any base, with all three common logs side by side and the antilog to check it.
Result
Logarithm: 3
Bases 10 and 2 are computed directly, so exact powers give exact answers.
- Natural log, ln
- 6.907755
- Common log, log₁₀
- 3
- Binary log, log₂
- 9.965784
- Check: base to the result
- 1,000
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How to use the Logarithm Calculator
- Enter the number, which must be greater than zero.
- [object Object]
- Read the logarithm, the three common logs side by side, and the check that the base raised to the result gives your number back.
The formula
log_b(x) = y means b^y = xlog_b(x) = ln x ÷ ln bThe number must be greater than zero, and the base must be positive and not 1.
The properties
Logarithms turn multiplication into addition, which is why they were used for calculation before electronic calculators:
log(a × b) = log a + log b log(a ÷ b) = log a − log b log(aⁿ) = n × log aExact powers stay exact
Working out log₁₀ 1000 as ln 1000 ÷ ln 10 gives 2.9999999999999996 on a computer, because each natural log is rounded. The page computes base-10 and base-2 logs directly, so exact powers of 10 and 2 give whole-number answers.
With your numbers
- log base 10 of 1,000 = 3
Quick reference
Bases 10 and 2 are computed directly, so exact powers give exact answers.
| Example | Logarithm |
|---|---|
| log₁₀ 1000 | 3 |
| log₂ 8 | 3 |
| ln e | 1 |
| log₇ 100 | 2.366589 |
| log of 1 | 0 |
| log of 0 |
Common questions about the Logarithm Calculator
- What is a logarithm?
- The power you raise the base to in order to get the number. log₁₀ 1000 = 3 because 10³ = 1000, and log₂ 8 = 3 because 2³ = 8. A logarithm answers the question "how many times do I multiply by the base?"
- Why can you not take the log of zero or a negative number?
- Because no power of a positive base is zero or negative. 10 raised to any power, however large or negative, is always a positive number; it gets closer to zero but never reaches it. So there is no exponent that gives 0 or −5, and the page says the logarithm is not defined.
- What are ln and log?
- ln is the natural logarithm, to base e ≈ 2.71828, which arises naturally in growth and decay. log usually means base 10, though in computing and higher mathematics it can mean base e or base 2, so the page names the base.
- How do I find a log in an unusual base?
- Divide the natural log of the number by the natural log of the base: log₇ 100 = ln 100 ÷ ln 7 = 2.36658932. This is the change-of-base formula, and it is how the page handles any base you enter.