mysterious equation
Well, this question, you are anxious about, is not so strange as it probably seems to you. The explanation of this circumstance is the followin:
1) Remember the main logarithmic identity, studied in the school course of algebra:
e^(ln(a)) == a, if a>0. Now let's apply this expression to your example:
2^(jwt) = e^(ln(2^(jwt))) = e^(iwt*ln(2)) = exp(jwt*ln(2)).
2) Now we simply adjust the Euler formula, which you mentioned, to exp(jwt*ln(2)) and get the following:
exp(jwt*ln(2)) = cos (wt*ln(2)) + j * sin(jwt*ln(2)).
So as you see again we obtained harmonic functions, which represent real and imaginary parts correspondingly. The 90 degrees phase shift preserves, the only difference - is that these functions are scaled (compressed, respectively the x-axis).
It's obvious, that any substitution of e doesn't change the sense.
With respect,
Dmitrij