Solving a general equation with sum of exponential

Question

Let $n>1$, and $K > 0$, consider the following equation

$$ \sum_{i=1}^n \alpha_i e^{\beta_i x}= K$$

where $\alpha_i, \beta_i \geq 0$ are known and $x \in \mathbb{R}$ unknown.

Is it possible to find an analytical solution $x$, if any, to the equation above?

Answer 1

I do not think there is any general analytic way to tackle such a problem. You can see it, if you put $y=e^x$ and then your equation becomes $$\sum_{i=1}^na_iy^{b_i}=K$$ which, even in the simplest case that $b_j$'s are natural numbers, where you get a polynomial equation, and if the degree is higher than $5$ there is no general closed-form algebraic solution by Abel's impossibility theorem.

For your original problem, in special cases, though, you may be able to work something out in terms of special functions.