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From: pruhs@uwvax.UUCP (Kirk Pruhs)
Newsgroups: net.puzzle
Subject: Re: x to the x to the x ...
Message-ID: <179@uwvax.UUCP>
Date: Thu, 25-Apr-85 15:13:59 EST
Article-I.D.: uwvax.179
Posted: Thu Apr 25 15:13:59 1985
Date-Received: Sat, 27-Apr-85 03:08:12 EST
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Organization: U of Wisconsin CS Dept
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> This one may be old, it's from an old MAA test.
> 
> x^x^x^x... = 2
> 
> the x's go on forever and the carots mean "power of", not "exclusive OR".
> What is the value of x?
> 

let y = x^x^x^...   then y = x^y.
y=2, now take log of both sides , ln y = ln2  or ln (x^y) =ln 2.
but ln(x^y) = y*ln x  and y=2. so we have 2*ln x =ln 2. Or if I can do this
right x= 2-e^2. Is this right?

Additional problems using the same idea:
in each of the following solve for x:

1. 2 = sqrt(x + sqrt(x + sqrt(x + (sqrt x + .......
2. 
    x = 1 +  1
            --------
	    1+ 1
               _______
	       1+ 1
		  _______
		      .
		       .
			.