Doublelog

Doublelog is commonly used to refer to the double logarithm, the result of applying a logarithm function twice to a positive input. In its typical form, it is written as log(log x), with the same base used for both applications. When the natural logarithm is used, the domain is x > 1, since ln x must be positive before taking its logarithm.

For x in the domain, the double logarithm is an increasing, concave function. If f(x) = ln(ln x), then f′(x) = 1/(x ln x) and f″(x) = −(ln x + 1)/(x^2 (ln x)^2), which is negative for x > 1. The range of f with natural logs is all real numbers: as x → 1+, ln x → 0+, so ln(ln x) → −∞; as x → ∞, ln x → ∞, so ln(ln x) → ∞. The exact form depends on the chosen base; using a fixed base b yields f(x) = log_b(log_b x).

Applications of the double logarithm arise in mathematical analysis and computer science where very slow growth is examined. It is often used to describe growth rates that exceed constants but remain slower than any power of x, and it is distinct from the iterated logarithm log*, which grows even more slowly.

In summary, doublelog typically denotes the double logarithm log(log x) with a consistent base, defined for x > 1 in the common natural-log convention, and characterized by slow, monotone growth.