Logarithm Calculator
Calculate logarithms in any base: natural (ln), base 10, base 2, or custom. See the change-of-base formula and antilog.
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About Logarithms
A logarithm answers the question: "To what power must the base be raised to produce x?" Formally, log_b(x) = y means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms transform multiplication into addition and exponentiation into multiplication, making them invaluable for calculations involving very large or very small numbers.
The three most common logarithm bases are: base e (natural logarithm, written ln), used in calculus and continuous growth models; base 10 (common logarithm, written log), used in pH, decibels, and the Richter scale; and base 2 (binary logarithm, written log₂), fundamental in computer science for measuring information in bits. Any base can be calculated using the change-of-base formula: log_b(x) = ln(x) / ln(b).
The antilogarithm (antilog) is the inverse operation: if log_b(x) = y, then the antilog is b^y = x. This verifies the computation. Key logarithm laws include: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(aⁿ) = n·log(a).
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For a related calculation, use Scientific Notation Calculator to convert between standard and scientific notation. Perform arithmetic on numbers in scientific notation. Alternatively, use Root Calculator to calculate the nth root of any number, including square roots, cube roots, and arbitrary roots with full working.
For the underlying method, read How to Use a Scientific Calculator: Functions, Angles, and Expression Order or Binary and Hexadecimal Explained: A Beginner's Guide.