AMC 12 2005a Problem 21

How many ordered triples of integers $(a,b,c)$, with $a \geq 2$, $b \geq 1$, and $c \geq 0$, satisfy both $\log_{a}b = c^{2005}$ and $a + b + c = 2005$?

$\textbf{(A) } 0 \textbf{(B) } 1 \textbf{(C) } 2 \textbf{(D) } 3 \textbf{(E) } 4$