The first-order theory over the reals as an ordered field with polynomials is decidable with doubly exponential complexity. However, if we additionally allow the exponential function, that is $e^x$ decidability is unknown (see here).
If we consider the existential fragment of first-order theory with polynomials the complexity is PSACE instead of doubly exponential (see here).
I am interested in results concerning decidability of existential first-order theory with polynomials extended with the function $a^x$, where $a$ is a rational number. I consider this not standard exponential function because of the application I have for this logic.