Incidentally, sure, we could just pick ^ to be our basic operator and define everything else in terms of that. Well, just as well, we could define everything in

Incidentally, sure, we could just pick ^ to be our basic operator and define everything else in terms of that. Well, just as well, we could define everything in terms of logarithm. Or everything in terms of roots! In fact, one might ask why you are so happy to define a logarithm operator v to undo ^ in one direction, but do not want ever to define and work with a root operator to undo ^ in the other direction. Is it because we can get away with saying a^(1/n) instead of n-th root of a? Well, just as well, why should we have a division operator? Instead of a/b, we can say a * b^(-1). Or perhaps we oughtn't have multiplication, or exponentiation with arbitrary bases. Perhaps we should just have addition, negation, natural logarithm, and the natural exponential. Then we define a * b as exp(ln(a) + ln(b)), define b^c as exp(c * ln(b)), etc. But, of course, just because F can be expressed in terms of G doesn't mean it's always convenient or pertinent to think of F that way; sometimes it's pertinent to think of F just qua F, as an atomic entity in its own right. And, thus... the proliferation of different functions people talk about. Though this does mean people then have to keep track of what all these different functions are defined as, and trace back through how that makes them relate. So... I don't know. I don't have all the answers. Yet.