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?

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, 
$$x = 10^{100}\quad or \quad x = 10^{1/100}$$
