logarithm problem - four tuple How many distinct four tuple (a,b,c,d) of rational numbers are there with $a\log_{10}2+b\log_{10}3+c\log_{10}5+d\log_{10}7=2005$
Can we proceed like this : 
Using $\log a +\log b = \log(ab)$ and $m\log a = \log a^m$
$\Rightarrow \log_{10}2^a \cdot 3^b \cdot 5^c \cdot 7^d = 2005$
Please guide how to proceed further..

Hint: $10^{2005}=2^a \cdot 3^b \cdot 5^c \cdot 7^d$
