logxln

LogxLn is not a standard mathematical term, but in informal contexts it may be used as a compact shorthand for the logarithm with base x applied to the natural logarithm of an argument. In formal terms, it can be written as log_x(ln(y)), the logarithm of ln(y) with base x.

The expression log_x(ln(y)) can be rewritten using the change-of-base formula as ln(ln(y)) / ln(x). This form makes the domain and properties clearer. For real-valued outputs, the inner ln(y) must be positive, so y must be greater than 1. In addition, the base x must satisfy x > 0 and x ≠ 1. Under these conditions, ln(x) is defined and the quotient is well-defined.

Behavior and properties follow from the rewritten form. The function is defined for y > 1 and x > 0, x ≠ 1. The derivative with respect to y is d/dy log_x(ln(y)) = 1 / (y · ln(x) · ln(y)). The value log_x(ln(y)) is zero precisely when ln(y) = 1, i.e., y = e. Near the boundary y → 1^+, ln(y) → 0^+, so log_x(ln(y)) tends to -∞ if x > 1 and to +∞ if 0 < x < 1. As y → ∞, ln(y) → ∞, so log_x(ln(y)) tends to +∞ for x > 1 and to -∞ for 0 < x < 1.

If x or y are outside the stated domain, the expression is not real-valued. In software, log_x(ln(y)) is typically computed via the natural logarithm: log_x(ln(y)) = ln(ln(y)) / ln(x). See also logarithms, natural logarithm, and change-of-base formulas.