Answer
Yes, it is possible.
Example: $2^x=0$
Work Step by Step
We are asked if it is possible for an exponential equation to have no solution. This is indeed possible. Consider:
$2^x=0$
We know that this equation has no solution because the graph of $y=2^x$ is always above the $x-axis$. The $y$ values get asymptotically close to zero, but never reach the value of $0$. Similarly, $2^x=-1$ would also have no solution, since $2^x$ is never negative.
