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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1891.0. "X^N = N^X ?" by CSOA1::LENNIG (Dave (N8JCX), MIG, @CYO) Thu Sep 01 1994 17:21

    Is there a general or special case solution (for x) to 
    
    		X^N = N^X
    
    I "see" that X=N satisfies the equation; however there is one (and only
    one?) other value for X that also satisfies it (verified experimentally)
    
    How do you derive an expression that will give you the other(s)?
    
    Dave
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1891.1can't doHERON::BLOMBERGTrapped inside the universeFri Sep 02 1994 09:397
    
    It would be equivalent to finding inverse(s) to ln(x)/x which
    I don't think you can in a closed form. Relates slightly to
    the problem of determining which number is the larger of e^pi and pi^e.
    
    /Ake
    
1891.2probably one answerAD::GRUNDMANNBillFri Sep 02 1994 11:4412
    How about this (haven't done much math in years)
    
    x^n = n^x
    ln(x^n) = ln(n^x)
    n * ln(x) = x * ln(n)
    n/ln(n) = x/ln(x)
    
    define f(n) = n/ln(n)
    we want to find two values where f(n1) = f(n2)
    
    If f(n) is monotonic, you can't do it. I think
    f(n) is monotonic, but how do you prove that?
1891.3AMCFAC::RABAHYdtn 471-5160, outside 1-810-347-5160Fri Sep 02 1994 14:081
    (2,4), (4,2), (-2,-4) and (-4,-2)
1891.4CSOA1::LENNIGDave (N8JCX), MIG, @CYOFri Sep 02 1994 14:3518
    I'm not sure what .3 is indicating...
    
    I also took things as far as ln(n)/n = ln(x)/x
    
    Interestingly, the _behaviour_ of ln(n)/n is sort of the same as the
    behaviour of the results I determined experimentally 
    
    The crossover points (X values) for various N
    	N			ln(n)/n
    	2	2	4	.151
    	3	2.4 	3	.159
    	4	2	4	.151
    	5	1.7	5	.140
    	6	1.6	6	.130
    
    I'm not sure what to make of the fact N=X isn't always the upper one.
    
    Dave
1891.5AMCFAC::RABAHYdtn 471-5160, outside 1-810-347-5160Fri Sep 02 1994 14:572
    .3 indicates some points that satisfy x**y = y**x other than the
    obvious x = y, (where x <> 0)
1891.6AMCFAC::RABAHYdtn 471-5160, outside 1-810-347-5160Fri Sep 02 1994 15:063
    re .4:
    
    Naturally, the crossover point is e.
1891.7topic pointer chainAUSSIE::GARSONachtentachtig kacheltjesMon Sep 05 1994 03:423
    re .*
    
    -> 1623 -> 512 (where it is analysed in depth)
1891.8CSOA1::LENNIGDave (N8JCX), MIG, @CYOTue Sep 06 1994 17:255
    Sorry - My DIR/TIT's didn't turn them up.
    
    And thanks - now I have to think about 512 some...
    
    Dave