Logxlny

Logxlny is a mathematical function defined as the logarithm with base x of the natural logarithm of y, written log_x(ln y). It requires x > 0 and x ≠ 1 as the base, and y > 1 to ensure the argument ln y is positive. The domain is thus x > 0, x ≠ 1, y > 1. Using the change-of-base formula, log_x(ln y) = ln(ln y) / ln x, where ln is the natural logarithm.

Examples help illustrate the definition. Let x = 2 and y = e^2. Then ln y = 2 and log_2(2) = 1. If y = e, ln y = 1 and log_x(1) = 0 for any valid base x.

Properties and calculus. For fixed y > 1, f(x,y) = log_x(ln y) is differentiable in x on (0,1) ∪ (1,∞) with partial derivative ∂f/∂x = - ln(ln y) / [ x (ln x)^2 ]. For fixed x with x > 0, x ≠ 1, the partial derivative with respect to y is ∂f/∂y = 1 / [ y ln x ln y ]. The function can take all real values as y varies (for any valid x), reflecting the range of the logarithm with a positive argument.

Notes and context. log_x(ln y) is a two-variable function that intertwines logarithms of different bases and is often used to illustrate change-of-base techniques, domain considerations, and multivariable differentiation in introductory and intermediate mathematics.