AMC 12 2005b Problem 17

How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there with

$a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005$?

$\mathrm{(A)}\ 0 \mathrm{(B)}\ 1 \mathrm{(C)}\ 17 \mathrm{(D)}\ 2004 \mathrm{(E)}\ \text{infinitely many}$