Suppose we have a vector
$$X=[x_1,x_2,\ldots,x_n],\quad x_i\in \mathbb{R} \text{ for } i=1,2,\ldots,n $$
Now if we have a formula $$f_X(x)=\log\left(\sum\limits_{i=1}^nx_i\right)$$ Is it possible to rewrite $f$ in any way, such that one can take the logarithm directly to the $x_i's$ ?
What you have here is the log of a sum of numbers. Unfortunately, there is no "simple" formula for rewriting this. Read the answer here for an example of what you can do to $log(a+b)$ for starters.