The original questioner asked if there was a version of Descartes' rule of signs for sums of exponential functions. The answer is yes.
Theorem. For $n\ge 0$, let $p_0>p_1>\cdots > p_n > 0$, and let $\alpha_j\ne 0$ be a real
number for $j = 0$, $1$, ... , $n$. Then the function
$$f(t) = \sum_{j=0}^n \alpha_j p_j^t$$
has no real zeros if $n=0$, and for $n\ge 1$ has at most as many zeros as there are among the $n$ pairs of successive constants
$$[\alpha_0,\alpha_1],\ [\alpha_1,\alpha_2],\ \cdots, [\alpha_{n-1},\alpha_n]$$
pairs where the two constants have opposite sign.
We note that the sum in the case $n=0$ has
plainly no zeros, because the sum consists of the single term $\alpha_0 p_0^t$, which is either strictly positive or strictly negative. We also note
that for arbitrary $n\ge 1$, the sum as no zeros if the entries in all the $n$ pairs of numbers have the same sign, because then all the terms in the sum are either
strictly positive or strictly negative. Equally elementary arguments show the result is true in the case $n=1$. The cases remaining are when
are $n\ge 2$ and where the number $m$ of the pairs of numbers having entries of opposite sign has $m\ge 1$. These cases we establish by induction on $n$.
Induction step: assume the theorem is true for an index $n\ge 1$, and all $m$ with $0<m<n$; and show it must therefore also be true for the index $n+1$, and for any $m$ in the as-yet unestablished cases where $1\le m < n+1$.
Examine the sum
$$f(t) = \sum_{j=0}^{n+1} \alpha_j p_j^t\ ,$$
and pull out some particular term, say the $k^{\rm th}$:
$$f(t) = \alpha_k p_k^t +\!\!\! \sum_{j=0,j\ne k}^{n+1} \alpha_j p_j^t\ .$$
Now $p_k>0$, so we can pull out a nonzero positive factor $p_k^t$ to get
$$f(t) = p_k^t \Big[\alpha_k + \!\!\!\sum_{j=0,j\ne k}^{n+1} \alpha_j ({p_j}/{p_k})^{t}\Big]\ .$$
Now we define
$$q_j = p_j/p_k\ .$$
Each constant $q_j$ is the ratio of two positive numbers, and so $q_j>0$. We also note that the sequence $q_j$ is strictly decreasing, $q_i>q_j$ for $i<j$.
We define
$$g(t) = \alpha_k+ \!\!\!\sum_{j=0,j\ne k}^{n+1} \alpha_j q_j^t\ ,$$
when we have $f(t) = p_k^t g(t)$, and consider the derivative
$$g'(t) = \sum_{j=0,j\ne k}^{n+1} \alpha_j\ln(q_j)\, q_j^t\ .$$
We define the nonzero constants
$$\beta_j = \alpha_j\ln q_j\ ,$$
when
$$g'(t) = \sum_{j=0,j\ne k}^{n+1}\beta_j q_j^t\ .$$
We see that $g'$ is a function that matches the assumptions of the theorem, with now $n$ terms in the sum.
The sign of the ratio $\alpha_j/\beta_j$ is negative for $j<k$, and positive for $j>k$, because we have $q_j>1$ for $j>k$
and we have $0<q_j<1$ for $j<k$, and because the function $\ln x$ is negative for $x<1$ and positive for $x>1$.
Examine an arbitrary pattern of signs for the constants $\alpha_j$ in the function $f$. For example, if $f$ were the sum of ten terms,
sign of $\alpha_j$ might vary with $j$ as
$$\pmatrix{
0&1&2&3&4&5&6&7&8&9\cr
+&+&-&-&+&-&-&+&-&+\cr}\ .$$
We have assumed $m>1$, and so there must be at least one pair of successive entries where the numbers in the pair
differ in sign. In particular, there has to be a smallest value of $j$ where the sign of $\alpha_j$ differs from the the sign of $\alpha_0$; and for
such a $j$ we must have $j\ge 1$. In the example given, we have $j=2$. Now in the expression for $g'$, choose $k={j-1}$;
since we know $j\ge 1$, we must have $k\ge 0$, so there must indeed be a constant $\alpha_k$ to correspond to that value of $k$.
The resulting sum for the function $g'$ will have the term with the constant $\beta_k$ omitted, and will have the signs of the $\beta_j$ for $j<k$ of opposite sign as $\alpha_j$, and
have all the signs of $\beta_j$ for $j>k$ of the same sign as $\alpha_j$. In the example given, the pattern of signs of the coefficents $\beta_j$ will
be
$$\pmatrix{
0&1&2&3&4&5&6&7&8&9\cr
-&.&-&-&+&-&-&+&-&+\cr}$$
The dot indicates that the coefficient $\beta_1$ does not appear in the sum that defines $g'$.
The key observation is that this process not only reduces the number of terms in the sum from the original $n+1$ in $f$ down to $n$ in $g'$,
but it also reduces the number of sign changes by one: if $f$ has $m$ such sign changes, then $g'$ will have $m-1$. Since $g'$ is otherwise an exponential sum in the proper form to apply the theorem,
by assumption $g'$ will have at most $m-1$ zeros.
By an application of Rolle's theorem, since $g'$ has at most $m-1$ zeros, we may conclude that its integral $g$ has at most $m$ zeros. But $f$ differs from $g$ only by being multiplied by the function $p_k^t$ that is never zero, and so $f$ must also have at most $m$ zeros. That completes the inductive step and proves the theorem.