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Thread: Challenge question

  1. #1
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    Challenge question

    Find x if x^{x^{x^{...}}}=2.

    I sent the answer to about 5 people since I have no idea how to figure out who all the moderators are.



    Moderator edit: Approved challenge question.
    Last edited by mr fantastic; Apr 19th 2010 at 09:38 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well clearly x^2 = x^{(x^{x^{x^{\dots}}})}=2, so x=\sqrt 2.
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  3. #3
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    Quote Originally Posted by Bruno J. View Post
    Well clearly x^2 = x^{(x^{x^{x^{\dots}}})}=2, so x=\sqrt 2.
    Correct, I did it like this.

    Find x if x^{x^{x^{...}}}=2.

    a=x^{x^{...}}=2

    ln(x^{x^{...}})=ln(2)

    x^{x^{...}}ln(x)=ln(2)

    aln(x)=ln(2)

    2ln(x)=ln(2)

    ln(x)=\frac{1}{2}ln(2)

    ln(x)=ln(\sqrt{2})

    x=\sqrt{2}
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  4. #4
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    which log rules did you use here?
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  5. #5
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    Quote Originally Posted by kandyfloss View Post
    which log rules did you use here?
    Exponential
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  6. #6
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    Quote Originally Posted by dwsmith View Post
    I sent the answer to about 5 people since I have no idea how to figure out who all the moderators are.
    The list of moderators is here:
    http://www.mathhelpforum.com/math-help/showgroups.php
    Don't be tricked by the "Moderators" heading - they are down at the bottom under "MHF Moderators".
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  7. #7
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    Hello, dwsmith!

    \text{Find }x \text{ if: }\:x^{x^{x^{\hdots}}}\:=\:2

    \text{We are given: }\;x^{x^{x^{\hdots}}} \:=\:2

    \text{Then: }\;x^{\left(x^{x^{\hdots}}\right)\;\leftarrow\;\te  xt{This is 2}}

    \text{The equation becomes: }\:x^2 \:=\:2

    \text{Therefore: }\:x \:=\:\sqrt{2}

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  8. #8
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    Quote Originally Posted by dwsmith View Post
    Correct, I did it like this.

    Find x if x^{x^{x^{...}}}=2.

    a=x^{x^{...}}=2

    ln(x^{x^{...}})=ln(2)

    x^{x^{...}}ln(x)=ln(2)

    aln(x)=ln(2)

    2ln(x)=ln(2)

    ln(x)=\frac{1}{2}ln(2)

    ln(x)=ln(\sqrt{2})

    x=\sqrt{2}
    I am completely supporting to his method as exponential method is perfect. I suggest you to this same method is not twisted and very easy to remember.
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  9. #9
    Ted
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    Quote Originally Posted by Soroban View Post
    \text{Then: }\;x^{\left(x^{x^{\hdots}}\right)\;\leftarrow\;\te  xt{This is 2}}
    Anyone could explain it for me !!
    why this is 2 !!
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  10. #10
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Ted View Post
    Anyone could explain it for me !!
    why this is 2 !!
    x^{x^{x^{\hdots}}} = 2

    so x^{\big{(}x^{x^{x^{\hdots}}}\big{)}} = x^2

    Since it's an infinite power tower of x's you can do that...
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  11. #11
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    Quote Originally Posted by Ted View Post
    Anyone could explain it for me !!
    why this is 2 !!
    x^{x^{x^{...}}}=2

    If we isolate an x, we still have an infinite x to the x which is stated as being equal to 2.
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