Logelog10x

Logelog10x denotes the natural logarithm of the common logarithm of x, expressed as f(x) = ln(log10(x)). It uses the natural logarithm (ln) and the base-10 logarithm (log10). The function is defined for x > 1, since log10(x) must be positive for the natural log to be defined. Equivalently, f(x) = ln( ln x / ln 10 ) = ln( ln x ) − ln( ln 10 ).

Basic properties and calculus: f is strictly increasing on (1, ∞), with derivative f'(x) = 1 / ( x ln x ). It is concave down on (1, ∞), since f''(x) = −(ln x + 1) / ( x^2 (ln x)^2 ) < 0 for x > 1. As x approaches 1 from the right, f(x) tends to −∞, indicating a vertical-like behavior at x = 1. As x → ∞, f(x) → ∞, so the function is unbounded above and has no horizontal asymptote.

Special values and relationships: For example, f(10) = ln(log10 10) = ln(1) = 0. The function’s range is all real numbers due to log10 x spanning (0, ∞) as x runs from 1 to ∞. The expression can be rewritten using the change-of-base formula to emphasize its form: log10 x = ln x / ln 10, so f(x) = ln( ln x ) − ln( ln 10 ).

Applications and related concepts: loge(log10 x) is a straightforward example of composing logarithms and is mainly of mathematical interest, illustrating how double-log transforms behave. It is related to the natural and base-10 logarithms and to common log transformations in data analysis, where the order of operations affects interpretation.

See also: natural logarithm, common logarithm, logarithmic transformation, change of base formula.