sinlog

Sinlog is a term used in mathematical contexts to denote the composition of the sine function with a logarithm, commonly written as sin(log x). By default, log refers to the natural logarithm, ln, so sinlog(x) = sin(ln x) for x > 0. The base of the logarithm can be varied, with sin(log_b x) = sin(ln x / ln b) representing a straightforward generalization.

Domain and basic properties: The real-valued function sinlog(x) is defined for positive x. Its range is [-1, 1]. Zeros occur when log x = kπ for integers k, i.e., at x = e^{kπ}. Unlike sin x, sinlog x is not periodic in x; as x increases, the oscillations become increasingly frequent because the argument ln x grows slowly but without bound.

Calculus and extensions: The derivative is d/dx sin(log x) = cos(log x) · (1/x). Higher derivatives follow from the chain and product rules, with the second derivative being -(sin(log x) + cos(log x)) / x^2. If the logarithm’s base b is used, the derivative becomes cos(log_b x) / (x ln b). The function can be extended to complex x via the complex logarithm, though this introduces branch cuts and multi-valued behavior.

Notes: Sinlog is not a universally standardized term and may appear informally to denote sin(log x) in examples or texts. It should not be confused with the log-sine integral, a distinct special function.