loge

Loge can refer to more than one concept, depending on context. In mathematics, loge denotes the natural logarithm, the logarithm to the base e. It is commonly written as log_e x or ln x. The natural logarithm is the inverse function of the exponential function e^x. Its domain is x > 0 and its range is all real numbers. The derivative is d/dx log_e x = 1/x for x > 0. An antiderivative is ∫ log_e x dx = x log_e x − x + C. The series expansion log_e(1 + x) = x − x^2/2 + x^3/3 − x^4/4 + … valid for |x| < 1. Key identities include e^{log_e x} = x and log_e(a^b) = b log_e a. Change of base can be written as log_a x = log_e x / log_e a. The base of the natural logarithm is the constant e, approximately 2.71828, and log_e x grows without bound as x increases, though more slowly than any positive power of x. In computing, loge is sometimes used interchangeably with ln or log with base e, depending on the language or calculator.

In theatre usage, a loge (or loge box) is a seating area—often a private or semi-private box near the stage. Loges originated as enclosed rooms or boxes that afforded a clearer view and some separation from the main seating. The term derives from the French loge, meaning a small room or box and is used in various venues to describe these privileged seats.

Overall, loge denotes either the mathematical natural logarithm or a type of theatre seating, with distinct meanings in mathematical and performing-arts contexts.