Demo that (e^(λ x))>(x^n) for large enough x

Module[{is=400,x,xc,ymin=-1,ymax=1},
Manipulate[
ymin=n Log[xmax/10];
ymax=n Log[xmax];
xc= Max[
Quiet[
x/.Solve[λ x==n Log[x],x],
{Solve::ifun}
]
];
(*Print[xc];*)
Column[{
Plot[{
λ x,
n Log[x]
},{x,0,xmax}
,PlotRange->{{0,xmax},{ymin,ymax}}
,PlotLegends->{"ln((E^(λ x)))","ln((x^n))"}
,ImageSize->is
,Epilog->{
PointSize[Large],
Point[{{xc,λ xc}}]
}
],
Plot[{
λ x,
n Log[x]
},{x,0,xmax}
,PlotRange->{{0,xmax},{0,10}}
,PlotLegends->{"ln((E^(λ x)))","ln((x^n))"}
,AspectRatio->0.2
,ImageSize->is
]
}],
{{λ,0.1},Appearance->"Labeled"},
{{n,10},1,1000, Appearance->"Labeled"},
{{xmax,10},Appearance->"Labeled"},
{xmax,Button["Find it",xmax=2xc]&}
]
]