Given a set $\mathbb{S}$ of sets of real numbers between 0 and 1, I know that solving $\min_{S\in\mathbb{S}}\prod_{s\in S}s$ could be reduced to $\min_{S\in\mathbb{S}}\sum_{s\in S}\ln(s)$, and this might be useful when you want to reduce a minimum probability problem to a minimum cost problem. Furthermore, we can reconstruct the original product value $\prod_{s\in S}s$ by computing the power of such a summation, i.e., $e^{\sum_{s\in S}\ln(s)}$.
Nevertheless, I am wondering whether it is possible to have the contrsirwise transformation, i.e., find the function $f$ such that $\min_{S\in\mathbb{S}}\sum_{s\in S}s$ could be reduced to $\min_{S\in\mathbb{S}}\prod_{s\in S}f(s)$ for each possible $\mathbb{S}$. Furthermore, I would be also interested in getting back the summation value, i.e., find $g$ such that for each $\mathbb{S}$:
$$g\left(\min_{S\in\mathbb{S}}\prod_{s\in S}f(s)\right)=\sum_{s\in S}s$$