When taking the continuum limit, as physicists often do, we turn all sums of the form $a\sum_n f(n)$ (where $a$ has the dimension of a length, for example) into integrals $\int dx \ f(x)$.
Suppose I have a product $\prod_n f(n)$ over the discrete variable $n$. What is the correct way to transform it in the continuous case? One way to do this could be passing to the logarithms: $$\prod_n f(n) = \exp\left(\sum_n\log f(n)\right) \to \exp\left(\int \ dx \log f(x)\right).$$ Is this how this is usually done?