What do you have in mind as a terse solution? I'm thinking as folllows: Let f(x) = x/log(x). Then f has a minimum at e - for example f'(x) = (log(x) - 1)/(l

What do you have in mind as a terse solution? I'm thinking as folllows: Let f(x) = x/log(x). Then f has a minimum at e - for example f'(x) = (log(x) - 1)/(log(x)^2), so f is decreasing for x < e and increasing for x > e. In particular f(e) < f(pi)-- that is, e / log(e) < pi / log(pi). Multiply through by log(e) log(pi) to get e log(pi) < pi log(e) and exponentiate both sides to get pi^e < e^pi.