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Let α\alpha α and β\beta β be two positive integers such that 43197<αβ<1777\dfrac { 43 }{ 197 } <\dfrac { \alpha }{ \beta } <\dfrac { 17 }{ 77 }19743<βα<7717 If the minimum possible value of β \beta β is xxx, find log2x\log _{ 2 }{ x } log2x.
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