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Topic: x^2 = 2^x and x^4 = 4^x
Replies: 5   Last Post: Oct 30, 2013 10:29 PM

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RGVickson@shaw.ca

Posts: 1,650
Registered: 12/1/07
Re: x^2 = 2^x and x^4 = 4^x
Posted: Oct 28, 2013 11:04 AM
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On Monday, October 28, 2013 6:30:46 AM UTC-7, James Ward wrote:
> I didn't see this curious fact mentioned in the wiki article (perhaps it is well known) http://en.wikipedia.org/wiki/Tetration
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> The two equations:
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> x^2 = 2^x and x^4 = 4^x,
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> both have 3 identical real solutions:
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> x = 2, 4, and -infinite power tower of (1/sqrt(2))
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> You can check the last in Wolfram Alpha using:
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> x = 1/sqrt(2),
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> y = -ProductLog((-log(x)))/(-log(x)),
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> z = 2^y - y^2
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> and
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> x = 1/sqrt(2),
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> y = -ProductLog((-log(x)))/(-log(x)),
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> z = 4^y - y^4


It follows by simple algebra: if x^2 = 2^x, then (x^2)^2 = (2^x)^2, so x^4 = (2^2)^x = 4^x. In fact, for any positive integer n we have that x^2 = 2^x implies x^(2n) = (2^n)^x. It is even true for positive non-integer values of n. No need for Wolfram Alpha here.



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