Logarithm Calculator
Calculate log₁₀, ln, log₂, and any custom base logarithm simultaneously. Enter any positive number.
log_b(x) = y means b^y = x. log (base 10): log(1000) = 3. ln (base e): ln(e) = 1. log₂: log₂(8) = 3. Change-of-base: log_b(x) = ln(x)/ln(b).
Quick values
Change of base: log_b(x) = ln(x) / ln(b) = log(x) / log(b)
log_b(xy) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) − log_b(y)
log_b(xⁿ) = n × log_b(x)
Logarithm Conventions by Country & Field
| Context / Country | When they write "log" | When they write "ln" |
|---|---|---|
| 🇺🇸 USA (engineering) | log₁₀ (common log) | Base-e (natural log) |
| 🌍 Pure mathematics | ln or log_e (natural log) | Same |
| 🇬🇧 UK (A-Level) | log₁₀ (unless context) | ln for natural log |
| 🇩🇪 Germany (Gymnasium) | lg for log₁₀, ln for natural | ln (same) |
| 💻 Computer science | log₂ (binary log) | ln for theoretical CS |
| ⚗️ Chemistry / pH | log₁₀ always | ln for rate equations |
| 💰 Finance | log₁₀ (general) | ln for continuous compounding |
Frequently Asked Questions
What does "log" mean on a calculator?
On most scientific calculators (TI-84, Casio fx), the "log" button is log₁₀. The "ln" button is the natural logarithm (base e). However, in European pure maths textbooks, "log" often means ln.
How is pH calculated using logarithms?
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L. pH 7 is neutral ([H⁺] = 10⁻⁷). Each pH unit is a 10× change in acidity — a pH scale used identically worldwide.
How is the Richter scale logarithmic?
Each whole number increase on the Richter scale means 10× more ground motion. A magnitude 7 earthquake is 10 times stronger than magnitude 6, and 100 times stronger than magnitude 5. Formula: M = log₁₀(A) − log₁₀(A₀).
What is the change-of-base formula?
log_b(x) = log(x)/log(b) = ln(x)/ln(b). This lets you calculate any base logarithm using only log₁₀ or ln. Example: log₅(125) = log(125)/log(5) = 2.0969/0.6990 = 3.