How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with \[a\log_{10} 2+b\log_{10} 3+c\log_{10} 5+d\log_{10} 7=2005?\]
Question:
How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with \[a\log_{10} 2+b\log_{10} 3+c\log_{10} 5+d\log_{10} 7=2005?\]
Answer:
Let's think step by step.

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How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with \[a\log_{10} 2+b\log_{10} 3+c\log_{10} 5+d\log_{10} 7=2005?\]
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