Calculate the logarithm of any number using natural log (ln), base 10, base 2, or a custom base. All four common logarithm forms shown simultaneously.
| Base | Notation | Name | Common Use |
|---|---|---|---|
| e ≈ 2.71828 | ln(x) | Natural log | Calculus, growth/decay |
| 10 | log(x) | Common log | pH, decibels, Richter scale |
| 2 | lb(x) or log₂(x) | Binary log | Computer science, bits |
| n | logn(x) | Custom base | Any logarithmic scale |
Change of Base Formula: logb(x) = ln(x) / ln(b)
"log" typically refers to the common logarithm with base 10, written log₁₀(x). "ln" is the natural logarithm with base e (Euler's number ≈ 2.71828), written logₑ(x). In mathematics and sciences, "log" often means natural log, while in engineering, "log" means base 10. Always check the context. This calculator makes the distinction clear by letting you choose the base explicitly.
Logarithms are only defined for positive numbers in the real number system. Since no real power of a positive base can produce a negative number or zero (b^y is always positive for positive b), log(0) and log(negative) are undefined in real numbers. In complex mathematics, logarithms of negative numbers are defined, but those results involve imaginary numbers and are outside the scope of standard calculators.