Understanding how the exponential function works is like putting the last piece in the middle of a puzzle: everything in calculus and algebra makes sense. Not only do a lot of things in real and complex analysis come from the very definition of Eulerâ€™s number and its consequent magic properties, but also things everybody uses in his mathematical life, like taking a real number-power of something, have their ultimate definition in \(e^x\).

For example, how do you calculate \(2^\sqrt{2}\) or \(3^\pi\) without knowing how the power operation with a non-rational exponent is defined?