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Math Help - What is x^x^x^x...

  1. #1
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    What is x^x^x^x...

    How would I find the solution, or simplify x^{x^{x^{x...}}}
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  2. #2
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    Re: What is x^x^x^x...

    Hi !

    it is an "infinite power tower"
    But your symbolism is ambiguous : it could means several different power towers. One have to use brakets to give a precise definition (in attachment)
    If the tower is finite (with a finit number of x), the related function of x can be formally expressed as the first derivative of a Generalized Sophomores Dream function : Formula 12:5 in the paper "The Sophomores Dream Function"
    Scribd
    Attached Thumbnails Attached Thumbnails What is x^x^x^x...-infinite-power-tower.jpg  
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    Re: What is x^x^x^x...

    Hello, jgv115!

    How would I find the solution, or simplify x^{x^{x^{x...}}}
    You can't . . .


    Let y \;=\;x^{x^{x\hdots}}

    Then: . y \:=\:x^y

    Take logs: . \ln(y) \:=\:\ln\left(x^y\right)  \quad\Rightarrow\quad \ln(y) \:=\:y\ln(x)

    And we have: . \ln(y) -y\ln(x) \:=\:0
    . . a transcendental equation in y.
    We cannot solve for y.


    By the way, JJ is incorrect.

    An exponential stack is always read "from the top down".


    For example, 4^{3^2} means: . 4^9 \:=\:262,\!144

    To change the order, parentheses are required: . (4^3)^2 \:=\:64^2 \:=\:4,\!096
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  4. #4
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    Re: What is x^x^x^x...

    Quote Originally Posted by Soroban View Post
    Hello, jgv115!

    You can't . . .


    Let y \;=\;x^{x^{x\hdots}}

    Then: . y \:=\:x^y

    Take logs: . \ln(y) \:=\:\ln\left(x^y\right)  \quad\Rightarrow\quad \ln(y) \:=\:y\ln(x)

    And we have: . \ln(y) -y\ln(x) \:=\:0
    . . a transcendental equation in y.
    We cannot solve for y.


    By the way, JJ is incorrect.

    An exponential stack is always read "from the top down".


    For example, 4^{3^2} means: . 4^9 \:=\:262,\!144

    To change the order, parentheses are required: . (4^3)^2 \:=\:64^2 \:=\:4,\!096
    Hi Soroban !

    you are right about the conventional hierarchy of operations.
    By the way, formal solving (for y) of the equation ln(y)-y*ln(x) is :
    y = -W(-ln(x))/ln(x) where W is the Lambert function.
    As others special functions, the LambertW function cannot be expressed as combination of a finite number of elementary functions.
    W(x) - Wolfram|Alpha
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