Logbcos

Logbcos is the notation for the function f(x) = log_b(cos x), the logarithm of the cosine of x with base b. In this form, log_b denotes the logarithm with base b, a function defined for positive bases b ≠ 1.

For real-valued outputs, the argument of the logarithm must be positive, so cos x > 0. The domain therefore consists of the portions of the real line where cos x is positive, namely x ∈ ⋃_{k∈Z} (−π/2 + 2kπ, π/2 + 2kπ). At points x = 2kπ, cos x = 1 and f(x) = log_b(1) = 0.

The derivative is f'(x) = (-sin x)/(cos x ln b) = -(tan x)/ln b, provided cos x ≠ 0. The function is 2π periodic on its domain, with the domain repeating every 2π.

The range depends on the base b. If b > 1 (ln b > 0), then cos x ∈ (0,1], so f(x) ∈ (−∞, 0]. If 0 < b < 1 (ln b < 0), the range is [0, ∞). As x approaches points where cos x → 0^+, f(x) → −∞ for b > 1 and f(x) → ∞ for 0 < b < 1.

Examples: f(0) = log_b(1) = 0, while f(π/3) = log_b(1/2) < 0 for b > 1. In programming contexts, logbcos may denote the same composition, often implemented as log(cos(x)) / log(b).

See also: logarithms, trigonometric functions, log_b of positive arguments.