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Power Equation

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PowerEquation

Consider solutions to the equation

x^y=y^x.
(1)

Real solutions are given by x=y for x,y>0, together with the solution of

(lny)/y=(lnx)/x,
(2)

which is given by

y={exp[-W_(-1)(-(lnx)/x)] for 1<x<e; exp[-W(-(lnx)/x)] for x>e,
(3)

where W_k(z) is the Lambert W-function. This function is illustrated above by the blue curve.

Rational parametric solutions are given by

x=(1+1/k)^k
(4)
y=(1+1/k)^(k+1)
(5)

for k=+/-1, +/-2, ... (Dunn 1980, Pickover 2002). These solutions are shown on the plot as red dots.

See also

Catalan's Conjecture, Lambert W-Function, Power

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References

Dunn, A. Mathematical Bafflers. New York: Dover, p. 213, 1980.Pickover, C. A. "The Gaps of Omicron." Ch. 23 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 53-54 and 265, 2002.

Referenced on Wolfram|Alpha

Power Equation

Cite this as:

Weisstein, Eric W. "Power Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PowerEquation.html

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