Floorlog23

Floorlog23 is the floor of the logarithm with base 23 for a positive real input. Formally, floorlog23(x) = floor(log_23(x)) for x > 0. The range consists of the nonnegative integers, and the function is undefined for x ≤ 0. It increases in a stepwise fashion, with jumps at powers of 23.

For integers, floorlog23(n) = k if and only if 23^k ≤ n < 23^{k+1}. Equivalently, floorlog23(n) gives the number of base-23 digit positions minus one. In this sense, the base-23 representation of n has floorlog23(n) + 1 digits. The function is monotone nondecreasing and changes value only when n crosses a power of 23.

Computation is straightforward with standard logarithms: floorlog23(x) = floor(ln x / ln 23) or floor(log10 x / log10 23). For integer inputs, an efficient approach is to compare n with successive powers of 23 or to use a logarithm function as above to obtain the result.

Examples: floorlog23(1) = 0, floorlog23(22) = 0, floorlog23(23) = 1, floorlog23(528) = 2, floorlog23(529) = 2, and floorlog23(12167) = 3, since 23^3 = 12167.

Applications of floorlog23 include determining the number of digits of a number in base 23, estimating sizes in algorithms that operate on base-23 representations, and related analyses in number theory and computer science. It is closely related to the general floor of a logarithm base b, with floorlog_b(n) equaling the digit-length minus one in base b.