What is greater e^pi or pi^e?
Let a^b >b^a, then blna>alnb, (lna)/a > (lnb)/b, Thus we graph the function (lnx)/x, We can see that this tends towards 0 as x tends towards infinity. We can also see that it is increasing from x=0 to a certain value of x. We can then find the maximum value of our function by finding the derivative. By using the product rule and setting our derivative to 0, we find x=e. Therefore (lne)/e>(lnb)/b for any b>0. Thus blne>elnb, e^b>b^e e^pi>pi^e
QED
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