Suppose $a_i,q_i,b_0$ are positive real numbers. I need to solve the following equation for $k$
$$\sum_i^d a_i \exp(-q_i k)=b_0$$
Is this a well-known problem? One my special cases has $a_i=q_i$
In my application $d> 20000$, $q_i\approx 0$. Wondering if equation structure makes it possible to solve faster than Newton's method.