To answer your question directly: The equation $6^x = 0$ has no solution for $x$, and therefore $\log_6 0$ is undefined.
To expand on the whole "no solution" vs. "undefined" thing, both "no solution" and "infinitely many solutions" (and in general anything other than "exactly one solution") mean that the expression representing the equation is undefined. For example, your example of $\frac 00$ can be represented as "the solution to $0x = 0$", which is any real or complex number (or in general any number in the field you're working in regardless of what that field is); therefore $\frac 00$ is undefined.
It is not always the case, however, that an undefined value will always stay that way. Take the case of the square roots. The equation $x^2 = 4$ has two roots, $2$ and $-2$, so technically $\sqrt{4}$ is "undefined". But because it suits our uses, we define the square root of a number to be its positive square root, and things work out.
In fact, the equation $x^2 + 1 = 0$ has no solution in the real numbers, so $\sqrt{-1}$ is also "undefined". But then we defined $i$, the imaginary unit, just to cover this case, and again, things worked out. Of course, by doing so we lost the property of ordering (Is $1$ or $i$ greater? There is no answer).
In general when you try and invent numbers to satisfy certain properties, you lose some properties that the previous system had. For example, extending the natural numbers to the integers to satisfy closure of subtraction means you lose well-ordering. Extending the integers to the rational numbers to satisfy division means you lose the existence of prime numbers, the division algorithm, and in general the possibility of numbers not being divisible by one another. Extending the rational numbers to the real numbers to satisfy the least-upper-bound property means that you lose finite representability (and maybe some other property that I don't know about), as some irrational numbers really do require an infinite, arbitrarily generated Cauchy sequence to define them.
In the case of your example of $\frac 00$, we could simply define $\frac 00$ to be $1$. However, by doing this, as you saw, we lose a lot of the properties that the real numbers give us, like multiplication being consistent. If we do this, then $1 = \frac 00 = \frac {2 \times 0}{0} = 2 \times \frac 00 = 2 \times 1 = 2$, meaning that things don't work out. In other words, by defining $\frac 00$, we lose too much other stuff for it to be useful. So we simply leave $\frac 00$ undefined and disallow it in most uses. (There are some cases in, say, calculus, where the form $\frac 00$ really is useful, but it's always in the context of limits and what value something takes as an expression approaches $\frac 00$.)
In short, the reason some things are undefined is simply because defining them causes trouble.
For more interesting reading on the subject of definitions and undefined numbers, try looking up the debate on whether $0^0 = 1$, or even the history of rational and irrational numbers. There have even been squabbles over whether negative numbers exist, and sometimes that makes for fun reading.
Integers are fractions, because a number is itself no matter how you write it.
A relevant section from Lockhart's A Mathematician's Lament:
In place of a natural problem context in which students can make decisions about what they want their words to mean, and what notions they wish to codify, they are instead subjected to an endless sequence of unmotivated and a priori “definitions.” The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?
Here's a relevant comic from SMBC:
Best Answer
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.