Transparent method
The formula
Worked example
Example inputs
How to interpret the result
A logarithm is really a question turned into an answer, what power gives this base this value, and losing sight of that makes the result look arbitrary. Log base 2 of 8 equals 3 because 2 cubed is 8, nothing more mysterious than that, and every logarithm here collapses back to the same exponent question with a different base and target.
Before you rely on it
What to check
Convert the answer back into exponent form, base raised to the result, and confirm it lands on the original value.
The common error
Where people go wrong with logarithm calculator
Confusing a change of base with a change of the underlying value. Log base 2 of 8 and log base 10 of 8 measure the identical quantity in two different unit systems, the way feet and meters measure the same distance differently.
Sensitivity evidence
How value changes the logarithm
| Value | Logarithm | base^result (check) |
|---|---|---|
| 80 | 1.9 | 80 |
| 90 | 2 | 90 |
| 100worked example | 2 | 100 |
| 110 | 2 | 110 |
| 120 | 2.1 | 120 |
Every input, tested
Which input moves the logarithm most
| Input | Tested from | To | Logarithm at each end | Swing |
|---|---|---|---|---|
| Base | 9 | 11 | 2.1 to 1.9 | 0.175 (8.8%) |
| Value | 90 | 110 | 2 to 2 | 0.087 (4.4%) |
Two variables at once
Logarithm by value and base
| Value \ Base | 8 | 10 | 12 |
|---|---|---|---|
| 80 | 2.1 | 1.9 | 1.8 |
| 90 | 2.2 | 2 | 1.8 |
| 100 | 2.2 | 2 | 1.9 |
| 110 | 2.3 | 2 | 1.9 |
| 120 | 2.3 | 2.1 | 1.9 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Value | 100 |
| Base | 10 |
| Logarithm | 2 |
| base^result (check) | 100 |
Where the inputs come from
Enter the value you want the logarithm of, which must be positive, and the base, which must be positive and not 1. A logarithm answers the question: to what power must the base be raised to get the value? Common bases are 10, for common logarithms, and e, about 2.718, for natural logarithms. The calculator uses the change-of-base formula, the natural log of the value divided by the natural log of the base.
When the result stops applying
A base of 1 is not allowed. Rounding errors matter for values near 1, where the result is close to zero. Different textbooks use log to mean base 10 or natural log, so check the convention before comparing answers.
What to compare it with
Solve time-to-grow problems with the exponential growth calculator, and see the rule of 72 for a shortcut. The exponent and root calculators cover the inverse operations, and the scientific notation page relates to powers of ten. Continuous compounding on the investing calculators uses natural logarithms.
People also ask
Frequently asked questions
What is a logarithm?
The exponent to which a base must be raised to produce a number. The base-10 logarithm of 1,000 is 3 because 10 cubed is 1,000.
What is the difference between log and ln?
Log usually means base 10 and ln means the natural logarithm with base e, though some fields use log for natural logs.
Why can't I take the logarithm of a negative number?
No real power of a positive base gives a negative result, so the logarithm is undefined among real numbers.