log2y

log2y is the logarithm of y with base 2. It is the exponent to which 2 must be raised to obtain y.

Defined for y > 0, log2 1 = 0 and log2 2 = 1. It satisfies log2(yz) = log2 y + log2 z and log2(y^k) = k log2 y. By the change-of-base formula, log2 y = ln y / ln 2.

The function is the inverse of the exponential function 2^x. Its range is all real numbers, while its domain is y > 0.

The derivative is d/dy log2 y = 1/(y ln 2), which is positive for y > 0, so log2 y is strictly increasing on its domain.

Applications and significance: log base 2 is common in computer science and information theory because doubling y corresponds to adding one bit of information. It appears in algorithm analysis, data compression, binary search, and the study of growth rates in binary systems. It also relates logs of different bases through simple constant factors, since log2 y = log10 y / log10 2 = ln y / ln 2.

Common pitfalls include taking the logarithm of non-positive values, confusing bases (for example, using base 10 instead of base 2 without adjusting), and forgetting the base-change factor when converting between bases.