log2N2

log2N2 refers to the base-2 logarithm of N squared, commonly written as log_2(N^2) or log2(N^2). It is defined for positive N, since real logarithms require positive arguments. The value can be interpreted as the amount of information or depth needed to address N^2 items in binary terms.

A key identity is log_2(N^2) = 2 log_2(N) for N > 0. This shows that log2N2 grows proportionally to the ordinary logarithm of N, with a constant factor of 2. Consequently, log2N2 = Θ(log N) as N grows large. The base-2 logarithm differs from natural or base-10 logarithms only by a multiplicative constant, since log_b x = log_k x / log_k b for any positive x and bases b, k.

Computationally, log2N2 can be obtained efficiently as 2 * log2(N). If N is large and the square N^2 is outside numerical ranges, evaluating the logarithm via the identity avoids computing N^2 directly. In many algorithmic contexts, log2N2 appears when the problem size scales with N^2, such as processing N-by-N matrices, indexing two-dimensional coordinates, or measuring information content in binary systems with N options per dimension.

Example: if N = 4, then N^2 = 16, and log2(16) = 4, which also equals 2 * log2(4) = 2 * 2 = 4.

See also: logarithm, base-2 logarithm, Big-O notation, complexity analysis.