The first step in performing a floating-point exponential operation is to scale things so that you're trying compute 2^(someInteger / powerOfTwo). This will easily reduce the problem to one of converting a value between 0 and 1, into a value between 1.00 and 1.99+. Let's suppose that we have a 32-bit integer and a scaling factor of 2^24. The upper 8 bits of that integer can easily yield the exponent for the result. As a rough approximation (to show how things are done), scale things so that the bottom 20 bits of the exponent will yield a value between 1.0 exactly and 1.04427378. Use bits 20-21 of the exponent to pick an entry from the table (1.0, 1.04427378, 1.09050773, 1.13878863) and multiply by that, and then use bits 22-23 to pick an entry from (1.00, 1.18920711, 1.41421356, 1.68179283) and multiply by that.
Note that this approach won't yield precise results, but it can achieve reasonably fast results with a reasonable amount of hardware. Using more lookup table steps can increase precision at the expense of time and hardware. Using larger lookup tables can increase precision at a given speed at the expense of hardware.