Enter a number and a base to get its logarithm, or use the antilog section to go the other way. Common bases (10, e and 2) are always shown alongside your custom base.
The logarithm logb(x) answers the question "what power do I raise b to, to get x?" So log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8. The number x must be positive, and the base must be positive and not equal to 1.
Calculators only have ln and log₁₀ built in, so any other base uses the change-of-base rule: logb(x) = ln(x) ÷ ln(b). For example, log₅(125) = ln 125 ÷ ln 5 = 4.8283 ÷ 1.6094 = 3.
The antilog reverses a logarithm: antilogb(y) = by. It is how you turn a pH back into a hydrogen-ion concentration (10−pH) or decibels back into a power ratio.
"log" on most calculators means base 10 (the common logarithm). "ln" is the natural logarithm, base e ≈ 2.71828, which appears in growth, decay and compound-interest formulas. In some maths and programming contexts, "log" means ln, so check which convention you are using.
No power of a positive base ever produces zero or a negative number, so the logarithm is undefined there (in real numbers). As x approaches 0 from above, the log heads towards negative infinity.
Use the change-of-base formula: log_b(x) = ln(x) / ln(b), or equally log₁₀(x) / log₁₀(b). This calculator does that for you and shows the working.
The antilog is the inverse of a logarithm: if log_b(x) = y, then x = b^y. The antilog of 2 in base 10 is 100.
Logarithmic scales measure things that span huge ranges: pH (acidity), decibels (loudness), the Richter and moment-magnitude scales (earthquakes), and star brightness. Logs also solve for time in growth and interest problems.
