TITLE: Is the sum of two exponential function can be equivalent to a third exponential function?
QUESTION [6 upvotes]: 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?

REPLY [13 votes]: No, it is not possible. Let $a\ne b$ and suppose
$$
Ae^{ax}+Be^{bx}=-Ce^{cx}
$$
for all $x$ (the minus sign is irrelevant, it just simplifies the argument).
If we use $x=0$, $x=1$ and $x=2$, we have the linear system in the unknowns $A$, $B$ and $C$:
$$
\begin{cases}
A+B+C=0 \\
Ae^a+Be^b+Ce^c=0 \\
Ae^{2a}+Be^{2b}+Ce^{2c}=0
\end{cases}
$$
whose matrix is
$$
\begin{bmatrix}
1 & 1 & 1 \\
e^a & e^b & e^c \\
e^{2a} & e^{2b} & e^{2c}
\end{bmatrix}
$$
which is a Vandermonde matrix, so its determinant is
$$
(e^c-e^a)(e^c-e^b)(e^b-e^a)
$$
so the system has just the trivial solution $A=B=C=0$, unless $c=a$ or $c=b$, in which cases we have $B=0$ or $A=0$, respectively.