logPxQx

Logpxqx is a mathematical shorthand used in some texts to denote the product of two logarithms of the same positive argument x with bases p and q: log_p(x) · log_q(x). The notation is not part of standard mathematical taxonomy but appears in a minority of literature and online discussions as a compact way to express this product. In many treatments, logpxqx is treated as a function of x parameterized by p and q.

By definition log_p(x) = ln(x)/ln(p) for p>0, p≠1, and x>0. Therefore logpxqx := log_p(x) log_q(x) = [ln x]^2 / [ln p ln q].

Domain: x>0; p>0, p≠1; q>0, q≠1. If x>1 and p>1, q>1, both logs are positive and the product is positive; if bases lie on opposite sides of 1 or x<1, the sign can vary.

Example: p=2, q=3, x=8. log_2(8)=3, log_3(8)=ln8/ln3≈2.0794/1.0986≈1.8928, so logpxqx ≈5.6784.

Applications: The expression can arise in theoretical analyses involving simultaneous logarithmic scaling with two bases, or in normalization schemes comparing growth rates across scales. It does not have a widespread standard use and is primarily of interest in pedagogical or exploratory contexts.

See also: Logarithm, Change of base formula, Logarithmic transformation.