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Next, we noted that the function f dened by f (x) = x is continuous on (0, ), so for any a > 0, the function x 1 dt is an antiderivative of f (x) on (0, ). We chose to name the particular one whose value is 0 at x = 1. We a t called this function the natural logarithm, and denoted it ln. Indeed, we might have taken this as the denition (once we knew such a function existed) and everything we know about ln can be proved from these two bits of information. So what do we know from this, and how did we get it?