[9question1]

"A base of 2 is useful because there are several small positive integers whose base-two logarithms are also integers." What? No! Base 2 is natural in exactly the same way that base e is natural, except for discrete domains instead of continuous domains. There is a unique family of functions for which the rate of change of the function is equal to the current value of the function everywhere. On discrete domains it's some scaled translation of 2^x, and on continuous domains it's some scaled translation of e^x. "Some scaled translation" here is accounting for the fact that the function is only uniquely exactly 2^x or e^x if we also add the constraint that f(0)=1.