2ln

2ln is a mathematical expression that denotes two times the natural logarithm of a variable, typically written as 2 ln(x). The natural logarithm, denoted by ln, is the logarithm with base e, where e is approximately 2.71828. In real-valued contexts, the natural logarithm is defined only for positive arguments, so 2 ln(x) is defined when x > 0.

The expression can be related to a single logarithm using log properties: for x > 0, 2 ln(x) = ln(x^2). This uses the rule ln(a^b) = b ln(a). Consequently, 2 ln(x) is the same as the natural logarithm of x squared, within its domain.

Key properties include the derivative and antiderivative. The derivative of 2 ln(x) with respect to x is 2/x for x > 0. An antiderivative is ∫ 2 ln(x) dx = 2x ln(x) − 2x + C, obtained by integration by parts.

Examples illustrate its use: 2 ln(3) ≈ 2.1972. To solve an equation like 2 ln(x) = c, one obtains x = e^(c/2). This follows from the inverse relationship between the natural logarithm and the exponential function.

In summary, 2ln is a simple multiple of the natural logarithm, defined for positive inputs, with standard algebraic and calculus properties inherited from the natural logarithm. It often appears in problems involving exponential growth, optimization, and integration.