sqrtLg

sqrtLg is a mathematical function defined as the square root of a logarithm of its argument. Formally, for a base b > 1, sqrtLg_b(x) = sqrt(lg_b(x)), where lg_b denotes the logarithm of x with base b. In many fields, the symbol lg denotes base-2 logarithm, so sqrtLg is often interpreted as sqrt(log base 2 of x). If the base is 10, it would be sqrt(log base 10 of x).

Domain and range: For real-valued outputs, the radicand must be nonnegative, so the domain is x >= 1. The range is [0, infinity). If complex values are allowed, the function extends to x > 0.

Derivative and monotonicity: The function is increasing on its real domain. Its derivative is d/dx sqrt(lg_b(x)) = 1 / (2 x ln b sqrt(lg_b(x))).

Asymptotics and comparison: Since lg_b(x) = ln x / ln b, sqrtLg_b(x) = sqrt(ln x / ln b). It grows without bound as x increases, but more slowly than any power x^c with c > 0, i.e., subpolynomial growth.

Examples: Example values for base b = 2: sqrtLg_2(2) = 1, sqrtLg_2(4) = sqrt(2) ≈ 1.414, sqrtLg_2(1) = 0. For base b = 10: sqrtLg_10(10) = 1, sqrtLg_10(100) = sqrt(2) ≈ 1.414.

Applications and notes: sqrtLg can appear in data transformations that aim to compress wide ranges or in analyses that follow a logarithmic scale with a subsequent square-root stabilization. Because notation varies, it is important to specify the base when using sqrtLg.