Today in class my teacher wrote something along the lines of:
$6^x = 0$
And proceed to heed a response from the class. A few people shouted undefined.
So the teacher then writes:
no solution $\therefore$ undefined
Now my question: Is undefined the same thing as no solution?
From what i understand about 'undefined' is not necessarily that the equation or expression has 'no solution' but rather that there can be infinitely many solutions. For example take $\frac{0}{0}$
Now if we have a line on the Cartesian plane, say, $ y=x$ then for every point on that line $\frac{y}{x} = 1$. But $(0;0)$ lies on the line so therefore $\frac{0}{0} = 1$
But now if we repeat that for the line $y = -x$ and follow the same argument we get $\frac{0}{0} = -1$ and so we can continue this argument for any line parsing through the origin with infinitely many gradients and thus infinitely many lines, and therefore we cannot 'define' $\frac{0}{0}$ by any one number.
Or at least that is what i was thinking when the question was written down.
Thanks!
Best Answer
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.