Could one argue that $10 \cdot 10 \cdot 10 \cdot 10 \cdots$ is equal to 0? The thing about this is, if we assign a variable: $$x=10 \cdot 10 \cdot 10 \cdot 10 \cdots$$ and then enclose all but one multiplicand in brackets (as multiplication is associative) and we get: $$x=10 \cdot (10 \cdot 10 \cdot 10 \cdot 10\cdots)$$ We can’t now see that the expression within the bracket is equal to $x$ so we get: $$x=10 \cdot x$$ The value of $x$ could clearly be shown as $0$ because $0=10\cdot 0$. This can’t be right because if we do the same thing with $1$ we can show that every number equals every other: $$x=1\cdot x$$$$4=1 \cdot 1 \cdot 1 \cdot 1 \cdots=3$$ 

My question is where is the flaw in this logic. Where does this method go wrong. Any insight would be helpful.

 A: First, we have to remind ourselves what it would mean for an infinite product to be equal to anything or to converge to something - much like the infinite summation, it would be the limit of the partial products, right? Well, the partial products for $\prod_{k=1}^\infty 10^k$ are $10, 10^2, 10^3, 10^4, 10^5$... obviously divergent. So, before I even address what you said: no, the product absolutely does not converge to anything.
Now, generally, this is a flaw not unlike with what Ramanujan ran into when showing that the summation $1+2+3+4+... = -1/12$. There's probably a proper formalization of this that someone else can elaborate on in the case of products, but I imagine the idea is the same.
Ramanujan's flaw was that the summation of the natural numbers is divergent. Thus, just "assigning" a variable to the value of the sum, i.e. saying $x = \sum_{k=1}^\infty k$, and then performing manipulations based on that to try to derive a value just is not kosher. The reason is because that summation is divergent - you can check the limit of the partial sums, and they visibly approach $\infty$.
Thus, I imagine an analogous idea holds here: you cannot say $x = \text{some infinite product}$ and perform manipulations as you did if that same product is divergent.
It's like a commenter said - to assume the product has a value and you can assign it to some constant $x$ is nonsense given it clearly has no value, and something reasonable cannot follow from nonsense.
Edit: As noted by a commenter, this is all under the assumptions of us working in the usual topology we normally work with. We could define an alternate topology in which these manipulations for this product make sense. So in a way, you're right - just not in our usual number system. :P
A: I guess the expression is 
$$ x=\lim_{k\to\infty} 10^k.$$
However, it doesn't make sense to "choose" a value for $x$. This limit diverges to $\infty$. So, your mistake is saying that $x$ is a real number satisfying $x=10x$. It turns out that $x"="\infty$. In particular, the expression  above is not a real number, because if $x\in \mathbb{R}$ then for any $r\in \mathbb{R}$, $x>r$. This is impossible.
So, this arithmetic is not meaningful, because arithmetic is not defined (in this context) for $\infty$.
