Ln3

Ln3, more commonly written as ln 3, denotes the natural logarithm of the number 3. It is the unique real number y such that e^y = 3, making it the inverse function value of the exponential function with base e. The natural logarithm is written with the abbreviation ln and is the basis of many mathematical formulas due to the special properties of the base e ≈ 2.71828.

Numerical value and basic properties: ln 3 is approximately 1.0986122886681098. The natural logarithm satisfies standard log rules: ln(ab) = ln a + ln b, ln(a^k) = k ln a, and ln e = 1. Its domain is the positive real numbers. ln 3 is irrational, a consequence of the transcendence of e.

Computation and relationships: ln 3 can be computed with calculators or numerical methods, including series expansions or numerical integration such as ∫_1^3 (1/t) dt. Change of base can express ln 3 in terms of common or binary logs, for example ln 3 = log10(3) × ln 10. The natural logarithm serves as the inverse of the exponential function e^x, a central concept in calculus and many applications.

Applications: ln 3 appears in problems involving growth and decay modeled with base e, solving equations where the unknown appears in an exponent, and in integrals of 1/x. It also features in physics, statistics, and information theory where logarithms quantify growth rates, entropy, and likelihoods.

Context: The natural logarithm is part of the broader family of logarithms and is denoted ln due to its association with the natural base e.