Ln1x

ln1x denotes the natural logarithm of 1+x. For real x, its domain is x > -1. The function maps (-1, ∞) to (-∞, ∞). It is strictly increasing and concave over its domain, with derivative 1/(1+x) and second derivative -1/(1+x)^2. It satisfies limits: as x approaches -1 from above, ln(1+x) tends to -∞; as x tends to ∞, ln(1+x) tends to ∞. The inverse function is exp(y) - 1, so x = e^y - 1.

Around x = 0, the Maclaurin series is ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ..., valid for -1 < x ≤ 1; the endpoint x = 1 yields ln 2, and the radius of convergence is 1.

Integral and identities: ∫ ln(1+x) dx = (1+x) ln(1+x) - x + C. For small x, ln(1+x) is approximately x. General inequalities from concavity include ln(1+x) ≤ x for x > -1, with equality at 0, and ln(1+x) ≥ x/(1+x).

Applications and notation: ln1p(x) is a common notation in computing for ln(1+x), used to improve numerical stability when x is small; it is widely implemented in mathematical libraries. The function is also central in calculus, asymptotics, and probability theory, where it serves as the logarithm of a shifted quantity and as the inverse of e^y − 1.