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But why is it "amazing" that e^{iπ} = -1? Is that any more special than e^{iπ} = 27 or e^{iπ} = -593.456?

Frankly, I always wondered what was so "natural" about "natural logaritms". The number "e" seems rather artificial to me, not even related to anything natural (as the number π is related to the circle). It seems to be invented just for the purpose of making ∫dx/x = ln x