Solve $\sum_{i=1}^n a_i\exp(-b_ix) = 1$ Problem. Solve $\displaystyle \sum_{i=1}^n a_i\exp(-b_ix) = 1$ where $n$ is a positive integer and $a_i, b_i (i=1,2,\dots,n)$ are positive constants.
I am not sure this can be solved analytically. The LHS is strictly decreasing and continuous from $+\infty$ to $-\infty$ so the solution exists and is unique.

Assuming that the $b_i$'s are rational numbers of irreducible denominators $d_i$, if we set
$$t:=\exp\left(-\frac x{\text{lcm}(d_i)}\right),$$
we get the polynomial equation
$$\sum_{i=0}^n a_i t^{n_i}-1=0$$ where $n_i=b_i\text{lcm}(d_i)$.
Hence this proves that the equation does not have an analytical solution in the general case.

If you consider the logarithm of the LHS (and look for its zeroes), it has two oblique asymptotes. For large positive $x$, it reduces to $\log a_{\min}-b_{\min}x$ and for large negative $x$, to $\log a_{\max}-b_{\max}x$ (the $\min$ and $\max$ indexes refer to $b$). This gives you an approximation of the root at the crossing point.

