Options[Plot]

Plot[{(x^10),(1.1^x)},{x,0,1000},PlotRange->{0,2000000000},PlotStyle-> {{Blue},{Red}}]

Solve[(b^x)==(x^a),x]

Solve[(x^10)==(1.1^x),x]

The following command takes a long time to finish

(* Reduce[(a^x)==(x^α),x] *)

In class I have discovered a condition for a and b under which the graphs of (x^a) and (b^x) will have exactly one common point. In that case these two graphs touch. Below I domostrate this discovery.

Plot[{(x^E),(E^x)},{x,0,10},PlotRange->{0,30},PlotStyle-> {{Blue},{Red}}]

a=2;Plot[{(x^a),(E^((a/E)x))},{x,0,10},PlotRange->{{0,10},{0,30}},PlotStyle-> {{Blue},{Red}}]

In class the command below did not work. It works fine in my office.

Manipulate[Show[Plot[Evaluate[{(x^a),(E^((a/E)x))}],{x,0,10},PlotRange->{{0,10},{0,30}},PlotStyle-> {{Blue},{Red}},Epilog->{Line[{{E,-1},{E,100}}]}]],{{a,1},.1,4}]