Could one generalize converting an infinite product to an infinite sum for the case of double products and double series as seen in $(1.1)$ and if possible how would one prove this ?
$(1.0)$
$$\log \prod_n s_n = \sum_n \log s_n$$
$(1.1)$ $$\log \left(\prod_n \prod_m s_{mn}\right) = \sum_n \sum_m \log (s_{mn})$$