Log base infinity

There is no \(\log_{\infty},\) be­cause \(\infty\) is not a real num­ber. Nev­er­the­less, the func­tion \(z\) defined as \(z(x) = 0\) for all \(x \in\) \(\mathbb R^+\) can pretty read­ily be in­ter­preted as \(\log_{\infty}\).

That is, \(z\) satis­fies all prop­er­ties of the ba­sic prop­er­ties of the log­a­r­ithm ex­cept for the one that says there ex­ists a \(b\) such that \(\log(b) = 1.\) In the case of \(\log_\infty,\) the log­a­r­ithm base in­finity claims “well, if you gave me a \(b\) that was large enough I might re­turn 1, but for all measly finite num­bers I re­turn 0.” In fact, if you’re feel­ing am­bi­tious, you can define \(\log_\infty\) to be a mul­ti­func­tion which al­lows in­finite in­puts, and define \(\log_\infty(\infty)\) to re­turn any pos­i­tive real num­ber that you’d like (1 in­cluded). This re­quires a few hijinks (like defin­ing \(\infty^0\) to also re­turn any num­ber that you’d like), but can be made to work and satisfy all the ba­sic log­a­r­ithm prop­er­ties (if you strate­gi­cally re-in­ter­pret some ‘$=$’ signs as \(\in\)″ signs).

The moral of the story is that func­tions that send ev­ery­thing to zero are al­most log­a­r­ithm func­tions, with the minor caveat that they ut­terly de­stroy all the in­tri­cate struc­ture that log­a­r­ithm func­tions tap into. (That’s what hap­pens when you choose “0” as your ar­bi­trary scal­ing fac­tor when tap­ping into the log lat­tice.)