Log calculator
Logarithms in any base: common log (10), natural log (e), binary log (2) or a base you choose. Plus the antilog to go back.
Antilog: by
A logarithm counts multiplications
Start at 1 and multiply by the base again and again. The logarithm is how many times you multiplied. Change the base and the count to see why log₂(32) = 5 and why each step of a log scale (decibels, pH, the Richter scale) is a multiplication, not an addition.
Two results fall straight out of this. Multiplying numbers adds their counts: log(a·b) = log a + log b. That is why slide rules and log tables turned multiplication into addition for 350 years after John Napier published logarithms in 1614.
Log laws
- log_b(xy) = log_b x + log_b y
- log_b(x/y) = log_b x − log_b y
- log_b(xⁿ) = n · log_b x
- log_b(x) = ln x / ln b (change of base)
- log_b(1) = 0, log_b(b) = 1
Common values
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 0.5 | −0.30103 | −0.693147 | −1 |
| 1 | 0 | 0 | 0 |
| 2 | 0.30103 | 0.693147 | 1 |
| 3 | 0.477121 | 1.09861 | 1.58496 |
| 5 | 0.69897 | 1.60944 | 2.32193 |
| 10 | 1 | 2.30259 | 3.32193 |
| 100 | 2 | 4.60517 | 6.64386 |
| 1024 | 3.0103 | 6.93147 | 10 |
Show me the steps
An AI tutor walks through the working for this exact result. The answer above is computed in your browser; the explanation is optional and is the only thing that leaves your device.
log base 10 of 1000 = 3
Questions people ask
What is a logarithm, in one sentence?
log_b(x) answers “what power do I raise b to, to get x?” So log₁₀(1000) = 3 because 10³ = 1000.
What is the difference between log and ln?
On calculators “log” means base 10 (the common logarithm) and “ln” means base e ≈ 2.71828 (the natural logarithm). In higher mathematics and many programming languages, log without a base often means ln: JavaScript’s Math.log and Python’s math.log are natural logs.
How do I calculate a log with a different base?
Use the change-of-base rule: log_b(x) = ln(x) / ln(b) = log(x) / log(b). This calculator does it for any positive base except 1.
Why can’t I take the log of zero or a negative number?
No real power of a positive base produces zero or a negative number, so the logarithm is undefined there. As x approaches 0, log(x) falls toward minus infinity.
What is an antilog?
The inverse of a logarithm: antilog_b(y) = b^y. The antilog of 3 in base 10 is 1000. Use the antilog box on this page, or 10ˣ and eˣ on the scientific calculator.