Is there a closed-form solution to the following equation? I would like to know if there is a closed-form solution for $x$ in the following equation. If there is no such form, how can you show this?
$$\frac{\sum{d_{i}^{x}\log(d_{i})}}{\sum{d_{i}^{x}}}=\frac{\sum{n_{i}\log(d_{i})}}{\sum{n_{i}}}$$
If it helps, we can assume that $d_i\ge1,n_i>0,x<0$

Note that it is notoriously difficult to prove that an expression can't be written in closed form in terms of elementary functions. But I certainly wouldn't wager on an equation involving the sum of arbitrary exponentials having one.
The left-hand side can be written as $\frac{d}{dt}\log \sum d_i^t\Bigg\vert_{t\to x}$, in case it somehow helps.
