Is it possible to approximate (or even find) a solution for the following equation: $$\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx0,$$
where $x\ge 0$ and integer, and the parameters $0<a<1$ and $u>0$, $v>0$.
It should be solved for $x$, can it be expressed in terms of the parameters?
Numerical methods show that such solution exists. For example, for $x=1$, $a=0.7$, $v=96.5543$ and $u=523$ the term value is $0.0246$.
Note: $x$ can be taken as a real number and then use the floor or ceiling functions to define the final expression.