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From: smp@gitpyr.UUCP (Scott M Pfeffer)
Newsgroups: net.puzzle
Subject: Re: x to the x to the x ... (SPOILED)
Message-ID: <380@gitpyr.UUCP>
Date: Fri, 10-May-85 02:15:32 EDT
Article-I.D.: gitpyr.380
Posted: Fri May 10 02:15:32 1985
Date-Received: Thu, 16-May-85 05:30:53 EDT
References: <748@whuxlm.UUCP> <351@gitpyr.UUCP> <821@mhuxt.UUCP>
Reply-To: smp@gitpyr.UUCP (Scott M Pfeffer)
Organization: Georgia Institute of Technology
Lines: 56

In article <821@mhuxt.UUCP> js2j@mhuxt.UUCP (sonntag) writes:
>    <solution deleted.  Presumably you've all seen it by now.>
>I couldn't figure out exactly what was wrong with the derivation, but
>Scott's answer:
>>              (1/2)
>>     6.  x = 2
>
>     Is easily demonstrated to be wrong, with the help of my trusty TI-58.
>Let x=2^(1/2).
>          x^x=1.632...
>	  x^x^x=2.000...
>	  x^x^x^x=2.665..., and the series just keeps on growing.  I suspect
>that the problem *has* no solution, and that x^x^x...=1 for  0<x<=1 and that
>x^x^x^x...approaches infinity for x>1.
>     I suppose that it is possible that there exists some non-real solution.
>-- 
>Jeff Sonntag
I too have see the error in my ways, but I cannot find any error in the deri-
vation. I did notice one interesting thing, however.
 
    In the "solution" I gave, the following appeared (more or less).

              (x^x^x^ ...)
    a)  log (x             ) = log (2)
           2                      2

    b)  (x^x^x^ ... ) log  (x)) = (1)
                         2

    This led to x = 2**(1/2). (for the fortran people left out there :-)
 
    Notice, though, that b) could have also been the following:

                           x
    b)  (x^x^x ... ) log (x  ) = (1)
                        2

    which would lead to "x to the x" equalling 2**(1/2), or
 

    c)  log (x**x) = log 2**(1/2)
    d)  xlogx      = (1/2)log2
    e)  xlogx      = (1/2)
    f)  logx       = 1/(2x)
    g)  If one plots  logx  and 1/(2x) on the positive half of an x-y
        graph, one finds that the only solution is somewhere between
        1 < x < 2, x not equal to 2**(1/2), try it.
 
   There must be something faulty that I am doing with logarithms and
   the "exponent" rule that my math teachers forgot to mention.  Does
   anyone out there know?  I'm lost.
 
          Scott Pfeffer
          {akgua, hplabs}!gatech!gitpyr!smp
          Georgia Institute of Technology
          Atlanta, Georgia