TITLE: Domain of $\log_2\log_3\log_2\log_3\log_2x$
QUESTION [5 upvotes]: I'm working on some math questions for a scholarship. I'm having a tough time as it seems to be tough for me. Please help me out

The function $\log_2\log_3\log_2\log_3\log_2x$ has the interval $x>?$ as its maximum domain on real numbers.

The key answer is $x>512$ and I have no idea how to go about with the answer.

REPLY [9 votes]: Work from the outside, peeling away the logarithms:
$$\log_2\log_3\log_2\log_3\log_2x\in\mathbb R\implies\log_3\log_2\log_3\log_2x>0$$
$$\log_3\log_2\log_3\log_2x>0\implies\log_2\log_3\log_2x>1$$
$$\log_2\log_3\log_2x>1\implies\log_3\log_2x>2$$
$$\log_3\log_2x>2\implies\log_2x>9$$
$$\log_2x>9\implies x>512$$
This works because the logarithm is a strictly increasing function.