Wikipedia:Reference desk/Archives/Mathematics/2007 July 18

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July 18

Question

Are there any other solutions for x^y = y^x other than y=x? --froth^t 19:44, 18 July 2007 (UTC)

Yes. For example 2^4 = 4^2 = 16. Try drawing the graph. It looks like one lobe of an hyperbola (but it isn't). --Trovatore 19:47, 18 July 2007 (UTC)

Here's how you can find more solutions :

$x^{y}=y^{x}\Leftrightarrow y\ln(x)=x\ln(y)\Leftrightarrow {\frac {\ln(y)}{y}}={\frac {\ln(x)}{x}}$

(With x and y both real)

This leads to the study of the function f(x) = ln(x)/x

For each value of a such that the line of equation f(x) = a cuts the curve f(x) = ln(x)/x in two points, there is a solution to the equation x^y = y^x, as this gives two different values of x for which ln(x)/x is the same.

--Xedi 20:35, 18 July 2007 (UTC)

It's worth noting that graphing $x^{y}=y^{x}$ can confuse some numerical methods a bit (the OS X Grapher application makes the curve a bit jaggy as it moves out, but panning/zooming eliminates the weird jagginess). I think, but I haven't proved, that the curved part of the graph intersects the straight line at (e,e). I think one way to show this is to calculate the derivatives of the graph and show that it is undefined at (e,e) but nowhere else on the line y = x. Donald Hosek 22:34, 18 July 2007 (UTC)

-- SGBailey 09:56, 19 July 2007 (UTC)

If you define the function u on the real numbers except 0 by:

$u(z)=\exp \left({\frac {z}{e^{z}-1}}\right)\,,$

all real-valued solutions in nonnegative reals of x^y = y^x with x ≠ y are given parametrically by:

$(x,y)=(u(z),u(-z)),~z\neq 0\,.$

The solution (2, 4) corresponds to z = 0.69314718... . The point 0 is a removable singularity; just define u(0) = e.

In the rationals, (x, y) = (–2, –4) is also a solution.  --Lambiam 23:49, 18 July 2007 (UTC)

Where does this solution come from ? --Xedi 17:44, 19 July 2007 (UTC)

From fiddling around with the equation x^y = y^x to get a parametric solution, and then some more fiddling to make it symmetric. I'm sorry, I have to confess it is OR, but readily verified.  --Lambiam 20:03, 19 July 2007 (UTC)

There is a Hungarian language article about this equation: Lóczy Lajos, Mikor kommutatív illetve asszociatív a hatványozás, Középiskolai Matematikai és Fizikai Lapok 2000/1 p. 7–16. However, that issue is not in the online archive, and it has no English abstract nor bibliography, so you can probably only use it if you speak Hungarian. – b_jonas 21:27, 23 July 2007 (UTC)

I think I've found an English translation: http://www.komal.hu/cikkek/loczy/powers/commpower.e.shtml. The article gives a parametric solution that is equivalent to mine, but not as symmetric. On the other hand, it considers the rational solutions and finds a whole family (for example, x = 117649/46656, y = 823543/279936), and also considers solutions of the "associative case", being the equation (x^y)^z = x^(yz).  --Lambiam 03:52, 24 July 2007 (UTC)

Oh, lucky you found it. I didn't look at the webpage because new articles almost never appear there nowdays and didn't think that this old article might be on. – b_jonas 10:44, 24 July 2007 (UTC)