If $r_i\in (0, 1), i = 1,\dots,N$, does there exists a unique $D > 0$ such that $\sum_{i=1}^N r_i^D = 1$?

Question

I am reading an article which states that if $r_i, i = 1,\dots,N$ are some Lipschitz constants, then there exists a unique positive number $D$ such that $\sum_{i=1}^N r_i^D = 1$. Granting the author the benefit of the doubt, let us assume that $N\geq 2$ and $r_i\in (0,1)$ for $i = 1,\dots,N$. In this case, how can you show the existence and uniqueness of $D$? I unfortunately cannot give any thoughts of my own since I have never really learned how to work with equations of the form $a^x + b^x = c$.