log2xy

Log2xy denotes the logarithm base 2 of the product of two positive numbers, written as log_2(xy). It is the exponent to which 2 must be raised to obtain the value xy.

Definition and domain. The expression log_2(xy) is defined for x > 0 and y > 0. If either x or y is zero or negative, the real-valued logarithm is undefined, though complex-valued logarithms can be considered in extended contexts.

Key properties. The product rule holds: log_2(xy) = log_2 x + log_2 y for x > 0 and y > 0. The power rule applies: log_2(x^k) = k log_2 x for any real k, with x > 0. Change of base can relate log_2 to natural or common logarithms: log_2 x = ln x / ln 2. These properties allow simplifying expressions involving products and powers.

Examples. log_2(8) = 3 because 2^3 = 8. log_2(3 × 5) = log_2 3 + log_2 5 ≈ 1.585 + 2.322 = 3.907. If x = 2 and y = 4, log_2(xy) = log_2 8 = 3.

Applications. log_2(xy) is used in information theory and computer science to measure data content in bits, in the analysis of algorithms with multiplicative growth, and in any context where binary scaling or binary representations are relevant.

Notation notes. To avoid ambiguity, log_2(xy) should be written with parentheses around the product. In programming, log2() is a common function name; the mathematical form remains log_2(xy).