INMO 2005 problem

Number Theory Level 4

Let \(\alpha \) and \(\beta \) be two positive integers such that \[\dfrac { 43 }{ 197 } <\dfrac { \alpha }{ \beta } <\dfrac { 17 }{ 77 }\] If the minimum possible value of \( \beta \) is \(x\), find \(\log _{ 2 }{ x } \).