Is the sum of two exponential function can be equivalent to a third exponential function? What will be the sum of two exponential functions $2\exp(4 x) + 3 \exp(5 x)$  equivalent to a third exponential function? Is it possible?

Just as goldenration commented, it is impossible to define the sum of two exponential functions as a third exponential function.
However, and this is just a long comment, in some cases, for some specific reasons and  at least locally, the sum of two exponential functions can be approximated by  a single exponential function and this can be very useful when you want to approximate solutions of equations.
Let us suppose that you want to find many many times a rough estimate of the  solution of equation $$2e^{4 x} + 3 e^{5 x}=k$$ $k$ being such that $0\leq x \leq 2$. A preliminary curve fit over that range would show you that over this specific range $$2e^{4 x} + 3 e^{5 x}\approx 3.99132 e^{4.90001 x}$$ So, suppose you want to solve approximately for $k=5000$; using the approximation and logarithms, this will immediately give $x \approx 1.45573$ while the exact solution would be $x \approx 1.45479$.
