I wanted to define a similar expression, and search for its properties: f ( n...

I wanted to define a similar expression, and search for its properties:
$f(n,x)={}^n\log x=\underset{n\text{ times}}{\underbrace{\log (\log (\log (\dots \log x)))}},n\in {\mathbb{N}}^{+},x\in \mathbb{R}$
just looking at how the domain explodes even when $n<8$ baffles me, as I obtained:
${\mathcal{D}}_{f(n\ge 2,x)}:x>{}^{n-2}e\wedge {\mathcal{D}}_{f(1,x)}:x>0\text{for example, }{\mathcal{D}}_{f(7,x)}:x\gtrsim {10}^{{10}^{{10}^{6.22}}}$
This may be not the only "strange" property of this particular function, but even if I think it can be exploited, I've never found any source (neither textbooks nor online); so in conclusion, is there any application of such function (or functions' class)?