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vichka [17]
3 years ago
11

Which function dominates x^p or p^x

Mathematics
1 answer:
maxonik [38]3 years ago
5 0
I'm assuming p\in\mathbb R is fixed. First note that x^p=p^x when x=p.

We can rewrite each expression as

x^p=e^{p\ln x}
p^x=e^{x\ln p}

Now to compare we only need to consider the exponents. It's easy to show that x>\ln x for all x>0, so it follows that x^p so long as x>p.
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