Power Equation

Author

Eric W. Weisstein
January 24, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/PowerEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PowerEquation.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Solution

Complexes

Reduce[((Log[y])/y)==(( Log[x])/x),{x,y}]//Timing

...hangs...

Rationals

Reduce[α^β==β^α,{α,β},Rationals]

FullSimplify[Subtract@@With[{α=(1+1/k)^k,β=(1+1/k)^(k+1)},
{α^β,β^α}],k>0]

FullSimplify[Subtract@@With[{α=(1+1/k)^k,β=(1+1/k)^(k+1)},
{α^β,β^α}],k≤-1]

Solve[{x==(1+1/k)^k,y==(1+1/k)^(k+1)},y,k]

Eliminate[{Log[x]==k Log[(1+1/k)],Log[y]==(k+1)Log[(1+1/k)]},k]//FullSimplify

Solve[Log[y]==((y Log[x])/x),y]

Solve[((Log[y])/y)==((Log[x])/x),y]

Plot[Piecewise[{{(E^(-ProductLog[-1,-((Log[x])/x)])),x<E}},(E^(-ProductLog[-((Log[x])/x)]))],{x,1.5,4.5},PlotStyle->Red,PlotRange->All]

Internal`DeactivateMessages[ParametricPlot[{(1+1/k)^k,(1+1/k)^(k+1)},{k,-300,300},MaxBend->.05,PlotStyle->Red,PlotRange->All]
]

Reals

Solve[(α^β)==(β^α),β]

Reduce[α^β==β^α,{α,β},Reals]

Internal`DeactivateMessages[Show[
Block[{$DisplayFunction=Identity},
{Plot[
Piecewise[{{(E^(-ProductLog[-1,-((Log[α])/α)])),1<α<E},{(E^(-ProductLog[-((Log[α])/α)])),α>E}}],{α,1,5},PlotStyle->{{Thickness[.01],Blue},Red}],
ListPlot[Table[{(1+1/k)^k,(1+1/k)^(k+1)},{k,-40,40}],PlotStyle->{Red,PointSize[.02]}]
}],PlotRange->{{1,5},{0,10}},AxesLabel->TraditionalForm/@{x,y}]
]