Log2x

Log2x, written log2(x) or log base 2 of x, is the logarithm with base 2. It yields the exponent to which 2 must be raised to obtain x, i.e., if y = log2(x) then 2^y = x. The function is defined for all positive x, with domain x > 0. Common values include log2(2) = 1, log2(8) = 3, and log2(0.5) = -1.

Key identities describe how log2 interacts with multiplication, powers, and reciprocals: log2(xy) = log2(x) + log2(y), log2(x^k) = k log2(x), and log2(1/x) = -log2(x). The change of base formula relates log2 to other bases: log2(x) = ln(x) / ln(2) = log10(x) / log10(2). As the inverse of the exponential function 2^x, it satisfies 2^{log2(x)} = x and log2(2^x) = x.

The range of log2 is the set of all real numbers, and its graph is a continuously increasing curve passing through (1, 0). As x approaches 0 from the right, log2(x) tends to negative infinity; as x grows, log2(x) grows without bound. The derivative is d/dx log2(x) = 1/(x ln 2) for x > 0, indicating a decreasing rate of growth as x increases.

Applications are common in computer science and information theory. Log2 is used to measure data size and complexity, such as determining the number of bits needed to represent a number or the asymptotic growth of algorithms (for example, logarithmic time or space). It also appears in analyses of binary search and binary encodings.