AMC 12 2004a Problem 17

Let $f$ be a function with the following properties:

(i) $f(1) = 1$, and

(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$.

What is the value of $f(2^{100})$?

$\text {(A)}\ 1 \text {(B)}\ 2^{99} \text {(C)}\ 2^{100} \text {(D)}\ 2^{4950} \text {(E)}\ 2^{9999}$