sqrtln

sqrtln refers to the real-valued function defined by f(x) = sqrt(ln x), where ln denotes the natural logarithm. For real outputs, the argument of the square root must be nonnegative, which occurs when x ≥ 1. For 0 < x < 1, ln x is negative and sqrt(ln x) is not real; complex-valued extensions are possible via branches of the square root.

On its real domain x ≥ 1, the function is increasing because ln x is increasing and the square root is increasing on [0, ∞). The derivative for x > 1 is f'(x) = 1 / (2 x sqrt(ln x)). At x = 1 the derivative is not finite, and the slope tends to +∞ from the right. The range of f on its real domain is [0, ∞).

An important property is the inverse relationship: if y = sqrt(ln x), then y^2 = ln x, so x = exp(y^2). Thus the inverse function is f^{-1}(y) = e^{y^2} for y ≥ 0. As x grows large, f(x) increases without bound, but slowly, roughly like sqrt(ln x).

Variants and notes: sqrtln can be generalized to sqrt(log_b x) = sqrt(ln x / ln b) for a base b > 0, b ≠ 1. Numerical evaluation requires domain checks (x ≥ 1 for real values) and careful handling near x = 1 due to the vertical tangent. In broader contexts, complex-valued extensions allow evaluation for x < 1 by choosing a branch of the logarithm and square root.

See also: natural logarithm, square root, logarithmic transformations, inverse functions.

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