Let $n$ be the product of all positive factors of $10^4$. Find $log_{10} n$ I can't see any relationship between the products of the positive factors of powers of 10.
The answer given here is 50, by the way, but I simply don't know where to start.

As @NL1992 pointed out, the key is write $10^4 = 2^4 \times 5^4$. 
Consider the factors which have no contribution from the "2". These are $5^0, 5^1, 5^2, 5^3$, and $5^4$. Their product is $5^{10}$.
Now, consider the factors which exactly one "2". These are $2 \times 5^0$, $2 \times 5^1$, $2 \times 5^2$, $2 \times 5^3$, and $2 \times 5^4$. Their product is $2^5 \times 5^{10}$.
Generalizing, you can see that if you multiply all factors which have a $2^k$ term, you get $2^{5k} \times 5^{10}$. 
So the final answer is $n = \prod_{k=0}^4 2^{5k}5^{10} = 2^{50} 5^{50}$, implying $\log_{10}(n) = 50$.
