lnln

lnln refers to the natural logarithm of the natural logarithm of a positive real number, written as ln(ln x). In real analysis, this function is defined only for x > 1, since ln x must be positive to take its logarithm.

As a function of x on (1, ∞), ln(ln x) is increasing, because its derivative f'(x) = 1/(x ln x) is positive for x > 1. It is also concave on (1, ∞), since the second derivative f''(x) = -(ln x + 1)/(x^2 (ln x)^2) is negative for x > 1. The function has a vertical kind of boundary at x → 1+, where ln x → 0+ and ln(ln x) → -∞. As x → ∞, ln x → ∞ and thus ln(ln x) → ∞, but very slowly.

In terms of growth, ln(ln x) grows much more slowly than ln x or any positive power of x. It is commonly described as a slowly increasing, double logarithmic function. For example, ln ln 10 ≈ ln(2.3026) ≈ 0.834, and ln ln e = ln 1 = 0.

The term lnln is often called the double logarithm and appears in contexts involving slow growth rates, such as certain asymptotic bounds in analysis and computer science. It is related to the broader family of iterated logarithm functions and is sometimes contrasted with the more rapidly growing logarithm and polynomial functions. In complex analysis, the real-valued definition is extended with care due to the multivalued nature of the logarithm.