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.
This free logarithm calculator helps you get accurate results instantly without needing to work through the math manually. Whether you need a quick estimate or a precise figure, having a reliable tool on hand saves time and reduces errors compared to manual calculation.
Small errors in math calculations can compound into significant problems. A 1% miscalculation in a budget or measurement can translate to hundreds or thousands of dollars in wasted resources, or result in having to redo work entirely. This calculator uses the standard formulas and methods to give you reliable results you can trust.
Is this calculator free? Yes — completely free with no account, login, or payment required. Use it as many times as you need.
How accurate are the results? This calculator uses the standard formulas for logarithm calculations. Results are accurate for typical use cases. For specialized or high-stakes applications, consult a professional.
Does it work on mobile? Yes. This calculator is fully responsive and works on smartphones, tablets, and desktops.