It really slows down though.
Log(n) indeed goes to infinity but if you numbered all the atoms in the universe and pass them through the function, the last one would be ~80.
That may be true, but that is a far cry from asymptotic
Mathematically, yes. For all real-world intents and purposes, it's bounded. I sometimes use this very example to illustrate the profound difference between the two which can be easily under-estimated. (That or log(log(n)) which also goes to infinity but is real-world bounded at <3)