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Calculate common, natural, binary, and custom base logarithms
Must be positive (x > 0)
A logarithm answers the question: "To what power must I raise the base to get this number?" For example, log₁₀(100) = 2 because 10² = 100.
log (common log) uses base 10, ln (natural log) uses base e ≈ 2.71828, and log₂ (binary log) uses base 2. They're all logarithms, just with different bases. ln is most common in calculus and science; log₂ is used in computer science.
In real numbers, you can't raise a positive base to any power and get a negative result. Therefore, logarithms of negative numbers are undefined in real numbers (they exist in complex numbers).
log_b(x) = ln(x) / ln(b) or log_b(x) = log₁₀(x) / log₁₀(b). This lets you calculate any logarithm using just natural or common logarithms, which are available on calculators.
e ≈ 2.71828 is Euler's number, a mathematical constant that appears naturally in growth and decay. The natural logarithm (ln) has special calculus properties: the derivative of ln(x) is simply 1/x.
Logarithms and exponentials are inverse operations. If y = b^x, then x = log_b(y). They "undo" each other: b^(log_b(x)) = x and log_b(b^x) = x.
Logarithms are used for pH (acidity), Richter scale (earthquakes), decibels (sound), information theory (bits), compound interest calculations, and anywhere exponential relationships exist.
Binary logarithms are fundamental in computing because computers use binary (base 2). log₂ tells you how many bits are needed to represent a number, and appears in algorithm complexity analysis (like binary search: O(log₂ n)).