y=e^x and y=x^2 have no intersections for x>0. y=e^x and y=x^3 have 2 intersections for x>0. The interesting question that follows is for what p : 2<p<3 does y=e^x and y=x^p have one intersection for x>o.

We need to find e^x=x^p.

take ln both sides and get x=p(lnx), which is p=x/lnx.

you will find that only when p=e, it has only one solution, by differentiation.

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