September 28th, 2014, 6:03 pm
Look at the logs of the two sides. Is pi > e * log(pi) ? In other words, is pi/log(pi) > e? Let's now differentiate x/log(x) and examine its value at pi. The derivative is (log(x) - 1)/log(x) ^ 2. The derivative is positive for x > e. Therefore pi/log(pi) > e/log(e) = e. Yes, pi > e * log(pi). This means that exp(pi) > pi ^ e.I'm not sure how to get Cuchulainn's more direct approach to work. When I tried that way, to analyse the derivative, I wanted to know the answer to the original problem, so I was going around in circles. (I've got a derivative, can't find the behaviour. I'm gonna try another approach. I've got a derivative, can't find the behaviour. I'm gonna try another approach. Will it go round in circles? Will it fly high like a bird up in the sky? etc.)CommodityQuant