lnlog10x

lnlog10x refers to the natural logarithm of the base-10 logarithm of x, written as ln(log10 x). In real numbers this is defined when log10 x > 0, which occurs for x > 1. Thus the domain is x > 1.

A convenient reformulation uses the identity log10 x = ln x / ln 10, so ln(log10 x) = ln(ln x) − ln(ln 10). This highlights that the function is a shifted version of ln(ln x): it equals ln(ln x) minus a constant, with the same domain restriction x > 1.

Key properties include:

- Monotonicity: the function is strictly increasing on (1, ∞). Its derivative is f'(x) = 1 / (x ln 10 · log10 x) = 1 / (x · ln 10 · (ln x / ln 10)) = 1 / (x ln x).

- Range: since log10 x > 0 for x > 1, ln(log10 x) takes all real values, ranging from −∞ as x → 1+ to ∞ as x → ∞.

- Behavior: as x approaches 1 from the right, log10 x → 0+, so ln(log10 x) → −∞. As x grows, log10 x grows without bound, so ln(log10 x) → ∞, albeit slowly.

Examples:

- x = 10 yields log10 10 = 1, so ln(log10 10) = ln(1) = 0.

- x = 100 yields log10 100 = 2, so ln(log10 100) = ln 2 ≈ 0.6931.

Applications and notes:

lnlog10x can arise in transformations where a double logarithmic scale is used or to linearize certain multiplicative models. Users should be mindful of the domain restriction and the distinction between natural logarithms and base-10 logarithms.