What's wrong with my aproach to solving this equation with multiple logarithms?

Question

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?

Answer 1

$$\log_{10}(x)=10\Rightarrow x=10^{10}$$ and $$\log_{10}(\log_{10}(x))=\log_{10}10=1\neq 4$$

Answer 2

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

Answer 3

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.

Answer 4

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