Can you simplify this otherwise tough problem??
The number of distinct \(4\)-tuples of rational numbers \((a,b,c,d)\) satisfying \[a \log 2+ b \log 3 + c \log 5 + d \log 7=2012\] is equal to \(k\). Enter your answer as the sum of \(k\) and all possible values of the \(4\)-tuples.
Details: Logarithm is taken to base \(10\).
Example: If \(k=2\) and the solutions are \((1,2,3,4), (5,6,7,8)\) then your answer should be \((1+2+3+4)+(5+6+7+8)+2.\)