log1p0

Log1p0 is not a standard mathematical symbol, but it is sometimes used to denote the evaluation of the log1p function at zero, that is log1p(0). Since log1p(x) is defined as ln(1+x), log1p(0) equals ln(1+0) which is ln(1) and therefore 0. The expression serves as a trivial reference point for the behavior of the log1p function.

The log1p function, commonly written as log1p(x), computes the natural logarithm of 1+x with numerical methods designed for accuracy when x is small. This avoids cancellation errors that can occur when computing ln(1+x) directly as ln(1+x) = ln(1+x) and 1+x is close to 1. The function is implemented in many programming languages and math libraries, such as C's log1p, Python's math.log1p, and Java's Math.log1p. For x near zero, log1p(x) has a stable series expansion: ln(1+x) = x − x^2/2 + x^3/3 − ...

At x = 0, log1p(0) evaluates to 0 exactly, and the derivative with respect to x at that point is 1/(1+0) = 1. In floating point arithmetic, 1+0 is represented exactly as 1, and log(1) is exactly 0, making log1p(0) an exact result in typical implementations. The special case log1p(0) serves as a simple verification point for numerical routines and for understanding the behavior of the log1p function.

See also: log1p, natural logarithm, numerical stability, Taylor series.