That's actually close to what happens with any well-founded infinitesimal theory. Bear with me a moment. Note that lots of functions have rates of growth betwe

That's actually close to what happens with any well-founded infinitesimal theory. Bear with me a moment. Note that lots of functions have rates of growth between x^0 and all x^a, so the rationale isn't great. Log(x) doesn't actually have to behave like any exponential. Worse yet, it's similarly pretty easy to prove that x^ε must be closer to 1 than any other real (for positive x). Log(x) seems curvier than a constant, so x^ε doesn't look like quite the right representation. That said, (x^ε-1)/ε _is_ actually closer to log(x) than any other real. The easiest way to get an intuitive feel for that is to rewrite x^ε as e^(ε log(x)) and examine the maclaurin series. There are some details in proving that works appropriately, but everything does actually fall into place. In some view then, x^ε really does have the same asymptomatic structure as log(x). They just differ by infinite constant factors rather than finite ones, so they definitely don't grow at the same rate in any classical sense (Big O, Big Theta, ...). Incidentally, exactly that same idea plays into one of the examples from TFA. Integrate x^(ε-1) with the ordinary power rule, and you'll find (1/ε)x^ε, or with that same maclaurin expansion (1/ε) + log(x) + O(ε). Admittedly the infinite constant looks a little odd (and I haven't taken care to prove that the results of that particular integration procedure are anything more than a happy accident), but that bears a striking resemblance to C + log(x).