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$~$ [positive_reals $~$\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 '[set_contains $~$\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.)