$a,b,c,d,f,g$ are parameters. $acf<0$. I tried to solve this:
$ae^{ax+b}+ce^{cx+d}+fe^{fx+g}=0$
But I cannot find any analytical solution. To make things worse, the wolfram alpha solver does not recognize this.
The sum of two exponential functions can be easily solved, though...
I have no idea how to proceed
$$ae^{ax+b}+ce^{cx+d}+fe^{fx+g}=0$$ They is a flaw in that question because several parameters are not independent. First the superfluous parameters will be eliminated. Divide by $fe^{fx+g}$ $$\frac{a}{f}e^{ax+b-fx-g}+\frac{c}{f}e^{cx+d-fx-g}+1=0$$ Let $\quad A=\frac{a}{f}e^{b-g}\quad;\quad B=\frac{c}{f}e^{d-g}\quad;\quad p=a-b\quad;\quad q=d-g$ $$Ae^{px}+Be^{qx}+1=0$$
Change of variable $\quad X=Be^{qx}\quad$ and let $\quad r=\frac{p}{q}\quad,\quad C=AB^{-r}$ $$C\;X^r+X+1=0$$ It is wellknown that the roots of this kind of equation in general cannot be expressed with a finite number of elementary functions.
Of course among the infinity of cases depending on $r$ and $C$ some particular are solvable in terms of elementary functions (for example polynomial equations of degree lower than 5 ) or in terms of special functions (for example polynomial of degree 5 in terms of Jacobi theta functions).
In the general case the roots have to be approximately computed thanks to numerical methods of calculus or on the form of series.