Suppose we have an equation of the form $$ \sum_{j=1}^n w_j \exp (x_jt)=0 $$
for a set of (real?) coefficients $w_j$ and $x_j$ and for $n>1$. Is it in general true that there is a (complex) $t$ that solves that equation?
The reason I ask is that I'm trying to find the poles of a function of the form $$ \log \sum_{j=1}^n w_j \exp (x_jt) $$
That, in turn, arises as the solution to a certain class of problems in network theory. I'm trying to understand when the Taylor series for that function has a finite radius of convergence