Minimize $\max(\sum_{i \in A}\log x_i, \sum_{i \in B} \frac{1}{x_i}\log \frac{1}{x_i})$ for $x_1,x_2, \dots, x_n$

Question

I want to minimize the following for the variables $x_1,x_2, \dots, x_n$: $$\max(\sum_{i \in A}\log x_i, \sum_{i \in B} \frac{1}{x_i}\log \frac{1}{x_i})$$ where $A,B \subset \{1,2, \dots, n\}$ with the condition that $A \cup B = \{1,2,\dots, n\}$ but the exact form of $A$ and $B$ are not known.

As $A$ and $B$ are unknown is it good to take $x_i=1$ for all $i$.