Log2xr

Log2xr is a notation sometimes used in mathematical and computational contexts to denote the binary (base-2) logarithm of a quantity indexed by r, frequently the r-th component of a vector x. Because it is not a single universally fixed function, its exact meaning can vary with discipline, but the most common interpretation is the logarithm base 2 of x_r, or log2(x_r).

Definition and notation. If x is a vector with components x_1, x_2, ..., then log2xr is typically written as log2(x_r) and defined for positive x_r. In matrix or array form, the operation can be applied elementwise, giving a new array where each component is log2(x_r) for the corresponding input component. The equivalent mathematical expression is log2xr = log2(x_r) = ln(x_r)/ln(2), valid for x_r > 0.

Properties. The function is monotone increasing on its domain, with a concave shape. It converts multiplicative scales into additive ones, since log2(a b) = log2(a) + log2(b) for positive a and b. When applied per component to a vector or array, these properties hold componentwise.

Calculation and handling of invalid inputs. In software and calculations, values with x_r ≤ 0 are undefined for the principal logarithm and typically yield NaN or an error, unless a specialized handling (such as complex logarithms) is used. Numerical implementations often provide vectorized support to compute log2xr across entire data structures.

Applications. Log2xr is used for data normalization and feature scaling, information-theoretic measures, and bit-length calculations (for example, determining the number of bits required to represent a positive integer). It also appears in analyses where quantities are naturally expressed on a binary logarithmic scale.

See also. Logarithm base 2, binary logarithm, log2, vector notation, componentwise operations.