coslog10

Coslog10 refers to the function f(x) = cos(log10 x), defined for x > 0. It combines the cosine function with a base-10 logarithm, yielding a smooth, bounded oscillatory curve on the positive real axis.

Key properties

- Domain and range: Domain is (0, ∞); the range is [-1, 1].

- Differentiability: The function is infinitely differentiable on its domain. Its derivative is f′(x) = -(1/(x ln 10)) sin(log10 x).

- Monotonicity: The function is not monotonic on any interval that contains more than a small portion of the x-axis; it oscillates indefinitely as x varies.

- Periodicity on a logarithmic scale: The function is periodic with respect to log10 x. Because log10(10^{2π} x) = log10 x + 2π, we have f(10^{2π} x) = f(x) for all x > 0. Thus the function repeats its values when x is multiplied by 10^{2π} ≈ 1.92 × 10^6, giving a multiplicative period.

- Zeros and extrema: Zeros occur when log10 x = π/2 + kπ, i.e., x = 10^{π/2 + kπ} for integers k. Maxima and minima occur at log10 x = kπ, giving x = 10^{kπ}.

Graphical and conceptual notes

- The amplitude remains constant at 1, but the oscillations persist indefinitely toward 0 and toward infinity.

- The function is often used as a simple model of log-periodic behavior, where phenomena exhibit regular cycles on a logarithmic scale.

Applications and context

- In mathematics, cos(log10 x) serves as a canonical example of a function with periodicity in the logarithmic domain.

- In applied contexts, it can illustrate log-scale periodicity in fields such as signal processing, physics, and certain financial models that analyze multiplicative processes.

See also: logarithm, cosine, log-periodic functions, log-scale analysis.