Logarithmic Equation

Algebra Level 4

If the solution of equation \({ 3 }^{ \log _{ n }{ x } }+5.{ x }^{ \log _{ n }{ 3 } }=3\quad (n>0,\quad n\neq 1)\) is equal to fractional part of \(\cfrac { { 3 }^{ 2m } }{ 8 } ,m\in \mathbb N\), and a polygon has \(n\) number of sides, then the ratio of number of diagonals of polygon to twice the number of sides is equal to \(\text{___________}\).

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