floorlogboriginal

floorlogboriginal is a mathematical operator defined for a positive real number x and a base b greater than 1, equal to floor(log_b x). In other words, it is the greatest integer m such that b raised to the mth power is at most x. The term combines elements of floor, logarithm, and base reference, and the word original in the name is used in contexts where multiple bases are considered to designate a canonical or default base.

In practice, floorlogboriginal(x, b) equals the usual floor of the logarithm with base b, and when b equals e it reduces to floor(ln x). The exact naming convention is not standardized across literature, but the functional meaning remains the same: it maps a positive input to the largest integer exponent that fits within x under base-b growth.

Examples illustrate the concept. floorlogboriginal(1000, 2) equals floor(log2 1000) = 9, since 2^9 ≤ 1000 < 2^10. floorlogboriginal(1000, 10) equals floor(log10 1000) = 3.

Properties of floorlogboriginal include that, for a fixed base, the function is nondecreasing in x and changes value only at powers of b, producing unit steps. For a fixed x > 1, increasing the base b yields a nonincreasing floorlogboriginal value. The function yields integer outputs and is piecewise constant as x varies.

Applications of floorlogboriginal appear in educational explanations of logarithms, analyses of digit counts in different bases, and algorithmic contexts where logarithmic scales or base-b partitioning provide a useful heuristic. See also: logarithm, floor function, base conversion, logarithmic scale.