A period in the sense of Kontsevitch-Zagier is a number expressible as an integral of an algebraic expression over an algebraic domain. Wang in https://arxiv.org/abs/1102.2273 introduced the notion of degree of a period as the minimal dimension of the algebraic domain. I would like to know if expressing a period of degree $d$ as a $d-1$ fold integral necessarily involves a logarithm in the integrand, the set of values of the logarithm at a given point corresponding to the set of equal subdivisions along the $d$-th axis of integration in the Riemann definition of an integral. This question arises from the Wikipedia article on Apéry's constant suggesting it is a period of degree $3$ (and thus a transcendental number), since it has a representation as a triple integral of a rational fraction, a representation as a double integral with a logarithm in the integrand and another one as a simple integral with a product of 2 logarithms in the integrand. So is there always a way to use $n<d$ logarithms in an $d-n$ fold integral to keep the value of the resulting integral constant?
Edit October 18th, 2023: the natural logarithm being a primitive of $x\mapsto 1/x$, it can be seen as an integral of a rational fraction and thus back up the considered conjecture. Maybe Fubini-Lebesgue's theorem applies here.