Logarithm Calculator
Base 10, natural, and base 2 logs for any positive value.
What it does
A logarithm asks which exponent on a base rebuilds the value: log10 of 1000 is 3 because 10^3 is 1000. The three standard bases split the work by field: 10 for scales and digits, e for calculus and growth, 2 for bits and doubling.
The tool above takes any positive value and returns all three logs at once, with the rebuild check in the note proving the first answer.
The default of 1000 reads 3 in base 10, 6.9078 natural, and 9.9658 in base 2. Powers of the base land exact integers: 100 gives exactly 2 in base 10, and 8 gives exactly 3 in base 2.
Definitions follow Khan Academy logarithm units with the same change of base rules in LibreTexts math references. Both restrict real logs to positive values, since no real power turns positive bases negative or zero.
How to use
- Enter a value above zero.
- Read log10 first for the familiar scale.
- Read ln for growth and calculus work.
- Read log2 for bits and halving counts.
- Raise the base to check any answer.
Formula
- 10
- Common log base for scales and orders.
- e
- 2.71828 natural base for growth math.
- 2
- Binary base for bits and doublings.
Zero and negatives are rejected because no real exponent fits them. Huge values cap at a quadrillion to keep four place precision honest.
Worked example
Example: logs of 1000
Value 1000.
- log10: 10^3 = 1000, so 3.
- ln: e^6.9078 = 1000, so 6.9078.
- log2: 2^9.9658 = 1000, so 9.9658.
- Check: 10^3 rebuilds 1000 exactly.
- Tool shows 3, 6.9078, 9.9658.
Reference table
| Value | log10 | log2 |
|---|---|---|
| 2 | 0.301 | 1 |
| 10 | 1 | 3.3219 |
| 100 | 2 | 6.6439 |
| 1000 | 3 | 9.9658 |
| 1024 | 3.0103 | 10 |
Limits
- Positive values only in real numbers.
- Four place display; tiny differences round together.
- Other bases need change of base by hand.
- Complex logs sit outside this page.
Common mistakes
- Logging zero or negatives and expecting answers.
- Mixing ln with log10 in formulas.
- Reading log2 of 1000 as 10 instead of 9.9658.
- Forgetting change of base divides, not subtracts.
- Rounding before raising to check.
Try it yourself
Quick checks. Answers sit tucked under each question.
log2 of 2?
1, since 2^1 = 2.
ln of e (2.71828)?
1, by definition of e.
log10 of 100?
2, since 10^2 = 100.
Questions
What is log10 of 1000?
3, because 10^3 = 1000.
ln vs log?
ln uses base e (2.71828) for growth; log here uses base 10 for scales.
Why log2 in computing?
Bits double per place, so log2 counts the bits a value needs.
Can I log other bases?
Divide: log_b(x) = ln(x) / ln(b) with this page values.
Why no log of zero?
No finite power of a positive base reaches zero or below.
What undoes a log?
Raising the base: 10^log10(x) rebuilds x.