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?

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
$$\log_{10}(\log_{10}(x))=-2\quad\Rightarrow\quad
  \log_{10}(x)=\frac{1}{100}\quad\Rightarrow\quad x=10^{1/100}\ .$$
