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Cooling problems proceed at a rate proportional to the temperature difference, so it proceeds most quickly when the temperature difference is greatest, and as the heat transfer proceeds the temperature difference reduces so the heat transfer slows. An arithmetic average only gives the right answer when the process proceeds at an even rate. When the maths is done in detail, and I have to admit I find it a bit too hard, it comes to integration of a 1/x term, which results in the natural log term, which properly accounts for the change in rate as the temperature difference decreases. The classic example is the cooling of a cup of coffee. Cooling proceeds at a rate proportional to the temperature difference between the coffee and the air. Not totally true with evaporation at a free surface, but not a bad approximation for a good cappacinno with a thick layer of good insulating foam on the surface. The temperature initially drops very quickly but then the temperature change becomes slower and slower. The cooling follows an exponential function which also involves that natural log function.