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2^x = x

  1. Apr 3, 2004 #1
    If 2^x = x

    then

    2^(2^x) = x

    also!

    2^2^(2^x) = x

    2^2^2^(2^x) = x

    2^2^2^2^2^2^2^ ...^(2^x)_n = x

     
  2. jcsd
  3. Apr 3, 2004 #2

    NateTG

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    Hmm.
    [tex]\frac{d}{dx} 2^x = \ln 2 2^x[/tex]
    which is monotone increasing and
    [tex]\frac{d}{dx} x = 1[/tex]
    which is constant, so if
    [tex]2^x > x[/tex]
    where
    [tex]\ln 2 \times 2^x = 1 \rightarrow x= -\log_2 ({\ln 2}) \approx 0.5[/tex]
    then the only solutions are imaginary.
    So I don't think there are any real solutions.
     
  4. Apr 3, 2004 #3
    Huh?

    If 2^x=x, then

    2^(2^x) does not equal x, it equals 2^x; you have to do the same thing to both sides, right??? I guess I don't understand your question then. It definitely does not have real solutions, because of the reason above.
     
  5. Apr 3, 2004 #4

    Hurkyl

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    Sure, 2^(2^x) equals 2^x... but what does 2^x equal?
     
  6. Apr 3, 2004 #5
    Ah i see, defined recursively
     
  7. Apr 3, 2004 #6
    x=-LambertW(-log2)/log2, i think...
     
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