Floorlog

Floorlog is the floor of a logarithm. For a positive number n and a base b greater than 1, floorlog_b(n) is the greatest integer k such that b raised to the power k is less than or equal to n. Equivalently, floorlog_b(n) equals floor of log base b of n. The base b determines the scale: base 2 is common in computing, base 10 in decimal digit counting, and other bases appear in data encoding or numeral systems.

The case of base 2 is especially notable: floorlog_2(n) corresponds to the position of the most significant one bit in the binary representation of n, counting from zero. It is related to the binary length of n, since for n > 0, floorlog_2(n) = bit_length(n) − 1, and bit_length(n) = floorlog_2(n) + 1.

Examples: floorlog_2(1) = 0, floorlog_2(2) = 1, floorlog_2(3) = 1, floorlog_2(4) = 2. For decimal base: floorlog_10(500) = 2 because 10^2 ≤ 500 < 10^3.

Computational notes: floorlog_b(n) can be computed by taking the floor of a logarithm, using floating-point arithmetic, or by integer methods that avoid floating-point computations. In many programming contexts, there are language features or library functions that yield log or digit counts, or that directly provide the position of the most significant bit.

Applications: floorlog is used to determine the number of digits of a number in a given base, to estimate time complexities of logarithmic algorithms, to size data structures, and in numeral systems and coding theory.

See also: logarithm, floor function, bit length, most significant bit, integer logarithm.