Log2

Log base 2, written as log2(x) or log_2(x), is the logarithm to the base 2. It is defined for all positive real x and equals the exponent y such that 2^y = x. The function is the inverse of the exponential function 2^x, and its values cover the entire set of real numbers as x runs over positive reals.

Key properties follow standard logarithm rules: log2(xy) = log2 x + log2 y, log2(x/y) = log2 x − log2 y, and log2(x^r) = r log2 x for any real r. The change-of-base formula allows converting between bases: log2 x = ln x / ln 2, where ln denotes the natural logarithm. Special values include log2(1) = 0, log2(2) = 1, and log2(1/2) = −1. The function is strictly increasing on (0, ∞) and has an unbounded range, approaching −∞ as x → 0+ and +∞ as x → ∞.

In computing and information theory, log2 is central because its units are bits, the basic measure of information. It underpins complexity estimates such as O(log n) for binary search, data encoding lengths, and entropy calculations in bits. Practical uses involve determining the number of times a value must be halved to reach a threshold, among other logarithmic scaling phenomena.