Complexity of exponential function
Here are some upper bounds.

By repeated squaring, the problem is in PSPACE.

There is a slightly better upper bound.  The problem is a special case of the BitSLP problem: Given a straight-line program starting from 0 and 1 with addition, subtraction and multiplication representing an integer N, and given i∈ℕ, decide whether the i-th bit (counting from the least significant bit) of the binary representation of N is 1.  The BitSLP problem is in the counting hierarchy (CH) [ABKM09].  (It is stated in [ABKM09] that it can be shown that the BitSLP problem is in PHPPPPPPPP.)

The membership to CH is often considered as an evidence that the problem is unlikely to be PSPACE-hard, because the equality CH=PSPACE implies that the counting hierarchy collapses.  However, I do not know how strong this evidence is considered to be.

As for the hardness, BitSLP is shown to be #P-hard in the same paper [ABKM09].  However, the proof there does not seem to imply any hardness of the language X in the question.

References

[ABKM09] Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen and Peter Bro Miltersen.  On the complexity of numerical analysis.  SIAM Journal on Computing, 38(5):1987–2006, Jan. 2009.  http://dx.doi.org/10.1137/070697926