If  ${\log _5}2,\,{\log _5}({2^x} - 3)$ and  ${\log _5}(\frac{{17}}{2} + {2^{x - 1}})$ are in $A.P.$ then the value of $x$ is :-

  • A

    $0$

  • B

    $-1$

  • C

    $3$

  • D

    $4$

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