Is there any way to solve equations like these? Is there any formula or method to solve equations like 

$ ax + b^x = c $

Where $ a, b, c $ are real numbers

I mean exact solutions not approximations 

Had any mathematician already worked on these?
The Maple command simplify(solve(a*x+b^x=c,x)); gives

$$\frac{c\ln(b)-a\,\mathrm{LambertW}\left(\dfrac{\ln(b)\,b^{c/a}}{a}\right)}{a\ln(b)}$$

So, there is a solution in terms of the Lambert W function, a special function. 
I don't know enough about it to explain it though but thought you might be interested.
Solving $f(x)=c$ involves intersection of curve $y=f(x)$ and a line $y=c$. This intersection can be computed with a graphs $(x,f(x))$ and $(x,c)$ in 2d space in such way that points $(x,y)$ are in the intersection if $y=f(x)=c$.

Your problem also has $a$ and $b$ so it requires some currying:
 $(a,b) \rightarrow (x) \rightarrow f(x,a,b)$. I.e. once you know values of $a$ and $b$, the equation solving problem can be attempted.