sqrtlog

Sqrtlog is a term used for the function f(x) = sqrt(log_b x), where log_b x denotes the logarithm of x with base b > 1. In many mathematical contexts the natural logarithm (base e) is standard, and in programming the base is often implicit. Because log_b x = ln x / ln b, changing the base scales the value by a constant factor.

Domain and range: For base b > 1, the logarithm is defined for x > 0, but the square root requires log_b x ≥ 0, which occurs when x ≥ 1. Therefore the natural domain is x ≥ 1, and the range is y ≥ 0.

Calculus and inverse: For x > 1, the derivative is f'(x) = 1 / (2 x ln b sqrt(log_b x)). The inverse relation is y = sqrt(log_b x) if and only if x = b^{y^2}, for y ≥ 0.

Growth and comparison: As x → ∞, f(x) grows without bound but very slowly. It grows faster than log log x but slower than log x. More precisely, f(x) = (log_b x)^{1/2} = (ln x)^{1/2} / (ln b)^{1/2} up to a constant factor depending on the base.

Examples: f_e(x) = sqrt(ln x). f_10(x) = sqrt(log10 x) = sqrt(ln x / ln 10).

Applications: Sqrtlog often appears in asymptotic analysis and complexity bounds as a slowly growing function, used to describe intermediate growth rates between logarithmic and polynomial terms. See also logarithm, square root, slowly growing functions.

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