Which is bigger, e to the pi or pi to the e?

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Both numbers are around 22 or 23. Which is larger, e to the power pi or pi to the power e? Find an argument that does not rely on decimal approximations, and that works for any two numbers in place of e and pi as long as one of them is e.

Accepted Answer

e to the pi is bigger (about 23.14 versus 22.46). Reduce to one function. Taking natural logarithms, e to the pi is bigger than pi to the e exactly when pi × ln e is bigger than e × ln pi, that is, when (ln e)/e is bigger than (ln pi)/pi. So compare the values of the function f(x) = (ln x)/x at x = e and x = pi. Where is (ln x)/x largest? Its derivative is (1 minus ln x)/x squared, which is positive for x below e and negative for x above e. So f rises until x = e and falls afterwards: its maximum is at x = e. Since pi is not e, f(pi) is strictly less than f(e), which gives (ln pi)/pi less tha…

Which is bigger, e to the pi or pi to the e?

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