ln

Ln, commonly written as ln, refers to the natural logarithm, the logarithm to the base e. It is the inverse of the exponential function with base e, exp(x) = e^x, and it maps positive real numbers to real numbers. For x > 0, ln x is the unique y such that e^y = x. The notation varies: many texts use ln, but some sources employ log_e x or, less often, capitalized forms like Ln.

Key identities include ln(1) = 0 and ln(e) = 1. For positive a and b, ln(ab) = ln a + ln b, and ln(a^b) = b ln a. The natural logarithm is strictly increasing on (0, ∞) and is concave down on that interval.

Differentiation and integration are fundamental: d/dx ln x = 1/x for x > 0. The integral ∫ (1/x) dx equals ln|x| + C, which, for x > 0, simplifies to ln x + C. A common series expansion around x = 0 is ln(1+x) = x − x^2/2 + x^3/3 − x^4/4 + … for |x| < 1, and related representations exist nearby 1.

Applications of the natural logarithm are widespread in mathematics, science, and engineering. It underpins continuous growth and decay models, appears in calculus and analysis, and serves as the inverse function to the exponential for solving equations involving exponential growth. In finance, continuous compounding uses e, as in A = P e^{rt}. In complex analysis, the natural logarithm extends with branch cuts, linking to broader topics in analytic functions.