Calculate logarithm with any base — log₁₀, natural log (ln), log₂, or any custom base. Also calculates antilog.
This tool calculates the logarithm of a number for any given base, including common logarithms (base 10) and natural logarithms (base e). Logarithms are the inverse of exponentiation and are widely used in science, engineering, and mathematics.
logₐ(x) = y means aʸ = x, where a is the base, x is the number, and y is the resulting logarithm. Common log uses base 10 (written as log(x)), and natural log uses base e ≈ 2.71828 (written as ln(x)).
Enter the number and the base you want to calculate the logarithm for, and the calculator returns the logarithm value.
Example 1: log₁₀(100) = 2, since 10² = 100.
Example 2: ln(e) = 1, since e¹ = e.
What's the difference between log and ln? "log" typically refers to the common logarithm (base 10), while "ln" refers to the natural logarithm (base e, approximately 2.71828), which appears frequently in calculus and natural growth/decay processes.
Why can't you take the log of a negative number or zero? Logarithms are only defined for positive numbers within real numbers, because no real exponent can raise a positive base to produce a negative result or zero.
What are logarithms used for in real life? Logarithms are used to measure earthquake magnitude (Richter scale), sound intensity (decibels), pH in chemistry, and to simplify calculations involving very large or small numbers, like in computer science and signal processing.
How do I change the base of a logarithm? Use the change of base formula: logₐ(x) = log(x) / log(a), allowing you to calculate a logarithm in any base using a calculator that only has common or natural log functions.
What is the relationship between logarithms and exponents? Logarithms and exponents are inverse operations — just as subtraction reverses addition, taking a logarithm reverses exponentiation, which is why logₐ(aˣ) = x.