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It can be determined that Solve[Gamma[x] == 2*Sqrt[Pi], x] yields a solution of x as x -> 3.5580029052596... What is being asked is: Is there a method that to convert the value of x numerical value to one represented by use of RealDigits ?

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Use N:

Solve[Gamma[x] == 2*Sqrt[Pi], x]
RealDigits[x /. N@%]
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    $\begingroup$ Solve returns the value with greater than MachinePrecision so use RealDigits[Solve[Gamma[x] == 2*Sqrt[Pi], x][[1, -1, -1, -1, -1]]] $\endgroup$
    – Bob Hanlon
    Commented Jan 14, 2017 at 6:47

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