e^x
= e^(a+b+c+d) (where abcd are the bytes of x)
= e^a * e^b * e^d * e*d
and then you make a lookup table for each of those e^a, e^b values.
(I fudged "a" here, where it's really more like "high byte << 24", but I think you just make your lookup table for e^a be a map from a => e^(a<<24), and similar for the other bytes.)
The first point lets you guess the order of magnitude of a result before doing the detailed calculation, and that is a great way to find errors.
The second one reveals why teachers get so annoyed when 9-digit answers are given to problems with 2-digit data.
I do wish some company would start producing cheap plastic slide rules again, so my students could try them out. The online ones are not as effective as actually seeing something in the real world, squinting your eyes to guess where the cursor lies between two lines on the rule. (Hm, I can get more digits on the left end of the rule than I can on the right ...)