lna

lnA denotes the natural logarithm of A, the logarithm with base e. It is defined for positive real numbers A > 0 and yields a real value. As the inverse of the exponential function e^x, ln and exp are inverse functions: e^(ln A) = A and ln(e^x) = x for all real x. The natural logarithm maps the positive real numbers onto the entire real line, with ln(1) = 0 and ln(e) = 1.

Key properties include ln(AB) = ln A + ln B and ln(A^k) = k ln A for A > 0, B > 0 and k real. The function is strictly increasing on (0, ∞). Its derivative is 1/A, and its integral is ∫ ln A dA = A ln A − A + C. The power series expansion around A = 1 is ln A = (A−1) − (A−1)^2/2 + (A−1)^3/3 − ..., valid for A near 1.

Common usages appear across mathematics, science, and finance to transform multiplicative relationships into additive ones and to measure relative changes. For example, ln 2 ≈ 0.6931, and ln e = 1. In computations involving different bases, the change-of-base formula relates natural logarithms to logarithms of other bases: ln A = log_k A · ln k.