What's wrong with my aproach to solving this equation with multiple logarithms? A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?"
My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$
However, this leads to an incorrect answer.  Instead, the solution posits I let $a=\log_{10}(x)$, and take the $\log$ base ten of both sides.  Why is my answer wrong?
 A: There's nothing wrong with the technique you've employed; you've made an assumption (namely $\log_{10}(x)=10$) and solved alfebraically for what $ x $ must be given the assumption.
$x $ having no solution here tells you that if your assumption was correct, then there is no solution to the equation. So either there's no solution to $ x $ or your assumption was wrong. 
The concept of imposing such assumptions and solving the outcome is, I think, a very important one to understand. Although the assumption you presented could have yielded a solution, it didn't. This doesn't mean that there is no solution at all, just that there is no solution which is consistent with the assumption you made. 
A: $$\log_{10}(x)=10\Rightarrow x=10^{10}$$ and $$\log_{10}(\log_{10}(x))=\log_{10}10=1\neq 4$$
A: Taking $\log_{10}$ of both sides and using the rule $\log_{10}(a^b)=b\log_{10}a$ gives
$$(\log_{10}(\log_{10}(x))(\log_{10}(\log_{10}(x))=\log_{10}10000=4\ ,$$
that is,
$$(\log_{10}(\log_{10}(x)))^2=4\ .$$
Hence
$$\log_{10}(\log_{10}(x))=2\quad\Rightarrow\quad
  \log_{10}(x)=100\quad\Rightarrow\quad x=10^{100}$$
or
$$\log_{10}(\log_{10}(x))=-2\quad\Rightarrow\quad
  \log_{10}(x)=\frac{1}{100}\quad\Rightarrow\quad x=10^{1/100}\ .$$
A: Your instincts are a fine attempt, but notice that no x can satisfy both conditions.
$$\log_{10}(x)=10 \quad and \quad\log_{10}(\log_{10}(x))=4$$
Your substitution would work, but it doesn't simplify the problem enough. 
It's probably easier to use this substitution:
$$Let\quad10^{10^a} = x$$
substitute: $$\log_{10}(10^{10^a})^{\log_{10}(\log_{10}(10^{10^a}))}= 10,000$$
simplify: $$(10^a)^a = 10^4$$
$$a = ±2\quad\Rightarrow\quad x=10^{10^{±2}}$$
so, 
$$x = 10^{100}\quad or \quad x = 10^{1/100}$$
