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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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jimward2@gmail.com

Posts: 7
Registered: 3/12/09
x^2 = 2^x and x^4 = 4^x
Posted: Oct 28, 2013 9:30 AM
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I didn't see this curious fact mentioned in the wiki article (perhaps it is well known) http://en.wikipedia.org/wiki/Tetration

The two equations:

x^2 = 2^x and x^4 = 4^x,

both have 3 identical real solutions:

x = 2, 4, and -infinite power tower of (1/sqrt(2))

You can check the last in Wolfram Alpha using:

x = 1/sqrt(2),
y = -ProductLog((-log(x)))/(-log(x)),
z = 2^y - y^2

and

x = 1/sqrt(2),
y = -ProductLog((-log(x)))/(-log(x)),
z = 4^y - y^4



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