ceilinglog2n

Ceilinglog2n refers to the mathematical function ceil(log2(n)), where log2 is the logarithm to base 2 and ceil is the ceiling function (rounding up to the nearest integer). The value is defined for n > 0; when n is a real number this yields an integer, and when n is a positive integer it yields a nonnegative integer.

Definition and interpretation: ceil(log2(n)) is the smallest integer m such that 2^m >= n. Equivalently, n lies in the interval (2^(m-1), 2^m] for m >= 0. This makes the function a natural measure of how many times you must double starting from 1 to reach or exceed n.

Examples: n = 1 gives 0, since 2^0 = 1. n = 2 gives 1, since 2^1 = 2. n = 3 gives 2, since 2^2 = 4 >= 3. n = 4 gives 2, since 2^2 = 4. n = 5 gives 3, since 2^3 = 8 >= 5.

Properties and relations: For positive integers n, if n is a power of two, ceil(log2(n)) equals log2(n). If n is not a power of two, ceil(log2(n)) equals floor(log2(n)) + 1. A common equivalent form for n > 1 is ceil(log2(n)) = floor(log2(n − 1)) + 1. These relations connect ceil(log2(n)) to the binary representation of n and the concept of bit-length.

Applications: The function appears in algorithm analysis and data structures, notably in determining the number of iterations in binary search, the minimum number of bits required to index up to n elements, and in various rounding and sizing calculations in computer science. It is also used in complexity bounds and in memory allocation contexts where powers of two are relevant.