cosln

cosln is a function defined by the expression cosln(x) = cos(ln x), where ln denotes the natural logarithm. Its domain is x > 0, since the natural logarithm is defined only for positive arguments. The range of cosln is [-1, 1], because the cosine function maps real numbers to this interval and ln x takes on all real values as x ranges over (0, ∞).

A key property of cosln is log-periodicity: the function is not periodic in x, but it is periodic in ln x with period 2π. Equivalently, cosln is invariant under multiplicative scaling x -> e^{2π} x, since ln(e^{2π} x) = 2π + ln x and cos(ln x + 2π) = cos(ln x). The zeros occur when ln x = π/2 + kπ, i.e., x = e^{π/2 + kπ} for integers k. Maxima and minima occur at ln x = 2kπ and ln x = (2k+1)π, giving f(x) = 1 and f(x) = -1 respectively.

Differentiation yields f'(x) = -(1/x) sin(ln x). Critical points arise when sin(ln x) = 0, i.e., ln x = kπ, so x = e^{kπ}.

As x approaches 0+ or ∞, cosln(x) continues to oscillate between -1 and 1, with the oscillation occurring on a logarithmic scale rather than a linear one. This makes cosln a useful illustrative example of a function that is dilationally invariant and log-periodic, often referenced in discussions of analysis and applications involving scale changes. In practical terms, cosln(x) can be computed as cos(ln(x)) for x > 0 in mathematical software and programming languages.