The Power of Logs!

Algebra Level 5

Let \(x,{ x }^{ \log _{ 10 }{ x } },{ y }^{ \log _{ 10 }{ y } },({ xy) }^{ \log _{ 10 }{ (xy) } }\quad \) be four consecutive terms of a geometric progression. Find the number of ordered pairs \((x,y)\) where \(x,y>0\) which satisfies the above condition.

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