Find original inputs $x$ and $^y$ for a given product, possible or not?

Question

$387,381,625,547,900,583,936$ is the product of this calculation $21\cdot2^{64}$.

If I only have the product and the multiplier $2$ (without the exponent) would it be possible to find the other inputs used to calculate that result, where those inputs were $x = 21$ and $y = 64$? If the answer is no, then might there be special cases where it could be possible? Or is it simply and unequivocally impossible to know for certain, no matter what, no special cases, period (and it's an absurd question)?

Solve for original inputs $x$ and $y:$

$387,381,625,547,900,583,936 = x\cdot2^{y}$

Answer 1

The answer, hinted at by the comments, is that there is not a unique choice unless you specify a way to normalize the form.

So all of the forms

$$(\underbrace{(20.1)2^{12-e}}_N)\times(2^{e})$$

are valid (where $e$ is any integer).

Normalizing would mean, for example, requiring that $0\leq N<2$. That would determine $N$ and $e$ uniquely ($N=1.25625$ and $e=16$).