Logarithm calculator
Compute logarithms in any base with the change-of-base formula, plus ln / log / log₂ shortcuts, exponential ⇄ logarithmic conversion, a common logarithm table and precision control.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
10
Logarithmic form：log₂(1024) = 10
The same relation b^x = n written two ways, rounded to the current precision
Derive the antilogarithm from base and exponent to verify both forms
| n | ln n | log n | log₂ n |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.69314718 | 0.30103 | 1 |
| 3 | 1.09861229 | 0.47712125 | 1.5849625 |
| 4 | 1.38629436 | 0.60205999 | 2 |
| 5 | 1.60943791 | 0.69897 | 2.32192809 |
| 10 | 2.30258509 | 1 | 3.32192809 |
| 100 | 4.60517019 | 2 | 6.64385619 |
| 1000 | 6.90775528 | 3 | 9.96578428 |
| 0.1 | -2.30258509 | -1 | -3.32192809 |
| 0.5 | -0.69314718 | -0.30103 | -1 |
| e | 1 | 0.43429448 | 1.44269504 |
What this tool does
- Compute logarithms in any base: when the base is not e, 10 or 2 the change-of-base formula log_b(n) = ln n / ln b is applied for you.
- Sanity-check algorithm complexity: 1024 gives log₂ = 10, the number of comparisons a binary search needs over 1024 items.
- Practise converting between exponential and logarithmic form: base plus argument yields both log_b(n) = x and b^x = n, and base plus exponent recovers the argument to prove they are inverses.
- Read off a common logarithm table for 1, 2, 3, 4, 5, 10, 100, 1000, 0.1, 0.5 and e in ln, lg and log₂, with precision adjustable from 0 to 12 decimals.
Example
Input
Argument n = 1024, base b = 2
Output
log₂(1024) = 10 Exponential form: 2^10 = 1024
Bases e, 10 and 2 go straight to Math.log / log10 / log2, avoiding binary residue such as Math.log(1000) / Math.log(10) returning 2.9999999999999996.
Frequently asked questions
Why are there limits on the base and the argument?
A logarithm needs n > 0 and b > 0 with b ≠ 1. Zero, negative numbers, base 1 and non-numeric input all raise an error rather than silently returning -Infinity or NaN.
Will the result carry floating-point error?
For e, 10 and 2 the matching Math.log implementation is used and results are typically exact; other bases go through the change-of-base formula and pick up a little error. Display rounds to your chosen precision with a tiny correction term, so values like 2.9999999999999996 do not survive as such.
How precise can the output be?
Between 0 and 12 decimal places. JavaScript numbers are IEEE 754 doubles with roughly 15–17 significant digits, so digits beyond 12 would be fictional — hence the cap.
What does "result outside the representable range" mean?
When deriving the argument as b^x, a result above about 1.8e308 becomes Infinity and a result below the smallest subnormal becomes 0; the tool reports an error instead of printing Infinity. Use an arbitrary-precision calculator for such numbers.
Is the computation uploaded anywhere?
No. Everything is computed locally in your browser with JavaScript, no network requests are made, and it works offline.
Keywords：logarithm对数loglnlglog2换底公式change of baseexponential form指数式