Logarithm Calculator (log, ln, log₂)
Type a number and pick a base to find the power the base must be raised to, with the working and a check.
log₁₀ 1000 =
3
log₁₀ 1000 = 3, because 10³ = 1000.
- Natural log (ln)
- ≈ 6.9077553
- Check
- 10³ = 1000
How logarithms work
A logarithm answers the question “to what power?”. log₁₀ 1000 = 3 because 10³ = 1000, and log₂ 32 = 5 because 2⁵ = 32. In general, log_b x = y means b^y = x, so logarithms undo exponents.
Answers aren’t always whole numbers. log₈ 4 = 2/3, because 8^(2/3) = 4 (the cube root of 8 is 2, and 2² = 4); this calculator finds such exact fractions. Most logs are irrational, such as log₂ 10 ≈ 3.321928. Calculators get them with the change-of-base rule, log_b x = ln x ÷ ln b, using the natural log ln (base e ≈ 2.71828).
Logs turn multiplication into addition, log(xy) = log x + log y, which is why they once powered slide rules and log tables. They also describe scales where each step is a multiple: pH, decibels and earthquake magnitudes all use logarithms. log₂ tells you how many bits you need: log₂ 1000 ≈ 9.97, so 10 bits count past 1,000.
log_b x = y ⇔ b^y = x · log_b x = ln x ÷ ln b
- John Napier published the first tables of logarithms in 1614, in Mirifici Logarithmorum Canonis Descriptio. Source: Wikipedia, Logarithm.
- pH is the negative base-10 logarithm of the hydrogen-ion activity, so each pH unit is a tenfold change: pH 3 is ten times more acidic than pH 4. Source: Wikipedia, pH.
Common logarithms in three bases
Base 10, base e (ln) and base 2 for numbers you meet often; exact answers are shown as such, the rest rounded.
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 0.001 | −3 | ≈ −6.90776 | ≈ −9.96578 |
| 0.01 | −2 | ≈ −4.60517 | ≈ −6.64386 |
| 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 |
| 8 | ≈ 0.90309 | ≈ 2.07944 | 3 |
| 10 | 1 | ≈ 2.30259 | ≈ 3.32193 |
| 16 | ≈ 1.20412 | ≈ 2.77259 | 4 |
Show all 16 rowsShow fewer
| 64 | ≈ 1.80618 | ≈ 4.15888 | 6 |
| 100 | 2 | ≈ 4.60517 | ≈ 6.64386 |
| 256 | ≈ 2.40824 | ≈ 5.54518 | 8 |
| 1,000 | 3 | ≈ 6.90776 | ≈ 9.96578 |
| 1,024 | ≈ 3.0103 | ≈ 6.93147 | 10 |
| 1,000,000 | 6 | ≈ 13.8155 | ≈ 19.9316 |
Frequently Asked Questions
What is the difference between log and ln?
log usually means base 10 (on calculators and in science) and ln means base e, the natural logarithm. In much of mathematics and programming, log on its own means ln, so check the convention.
Why is the log of a negative number or zero undefined?
No power of a positive base is zero or negative: b^y is always above 0. So there is no real y with 10^y = −5 or 10^y = 0.
What is an antilog?
The inverse of a log: the antilog of y in base 10 is 10^y. If log₁₀ x = 2.5, then x = 10^2.5 ≈ 316.23. Use the exponent calculator for it.