What is the inverse function of $\alpha\mathrm{e}^{\beta x}+\gamma\mathrm{e}^{\delta x}$? I need to solve this equation for $x$: $\alpha\mathrm{e}^{\beta x}+\gamma\mathrm{e}^{\delta x}=\epsilon$
($\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ are real constants)
I'm only interested in real solutions.
Is such an equation solvable? Even Wolfram|Alpha refuses to solve trivialized versions of this equation: $\mathrm{e}^{\alpha x}+\mathrm{e}^{\beta x}=a$ or $\alpha^x+\beta^x=a$
 A: Letting $z = \alpha e^{\beta x}/\epsilon$, and assuming the parameters are positive, the equation becomes 
$$ z + \dfrac{\gamma \epsilon^{\delta/\beta-1}}{\alpha^{\delta/\beta}} z^{\delta/\beta} - 1 = 0$$
which I'll write as
$$ z + c z^p - 1 = 0 $$
This has a series solution in powers of $c$, that should converge for small $|c|$:
$$\eqalign{z &= \sum_{n=0}^\infty \dfrac{((-c)^n}{n!} \prod_{j=0}^{n-2} (np - j)\cr
&= 1 - c + p c^2 - \dfrac{3p(3p-1)}{6} c^3 + \dfrac{(4p)(4p-1)(4p-2)}{24} c^4 + \ldots}$$
A: It isn't possible to find a general solution. Letting $y = e^x$, we have
$$\alpha y^\beta + \gamma y^\delta = \epsilon$$
Even for some integer values of $\alpha, ...,\epsilon$, this could give us polynomials whose solutions cannot be expressed in terms of ordinary operations (addition, subtraction, multiplication, division, and roots). For example, the roots of
$$y^5-4y=1$$
can't be expressed in terms of ordinary mathematical operations, since it's irreducible and it has three real roots and two complex roots, and thus has Galois group $S_5$. And so if we can't find a solution for this special case, certainly there won't be a general solution. 
