lnXi

lnxi denotes the natural logarithm of xi, a notation commonly used in mathematics and related fields where xi represents a positive real variable or a dimensionless quantity. The natural logarithm, written ln, is the logarithm with base e, where e is the Euler number. For real xi > 0, ln xi is a real-valued function that increases monotonically from −∞ as xi approaches 0 from the right to ∞ as xi grows without bound. It is undefined for xi ≤ 0 in the real sense; complex extensions exist and introduce branch cuts.

Key identities include ln(xy) = ln x + ln y, ln(x^a) = a ln x, and the inverse relation e^{ln x} = x for x > 0. The derivative d/dxi ln xi = 1/xi, and an antiderivative ∫ ln xi dxi = xi ln xi − xi + C. The natural logarithm converts multiplicative changes into additive ones and is widely used to model growth and to simplify products in algebra and calculus.

In practice, xi often represents a positive quantity such as a ratio, a probability-density parameter, or a dimensionless measure, and ln xi appears in expressions like likelihoods, entropy, and growth models. In programming and typography, ln is used interchangeably with log when the base is e, though some software uses log for the natural logarithm and log10 for base-10.

Complex-valued ln is multivalued and requires a chosen branch; the principal value is defined by selecting a branch cut typically along the negative real axis, with the result differing by multiples of 2πi.