Domain of $\log_2\log_3\log_2\log_3\log_2x$

Question

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.

Answer 1

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.