Domain of a composite function

Question

I was given the question:

Find the domain of the function $f(x)=\ln(\ln(\ln x))$

I found the answer by inspection:

$\qquad D(\ln x)=(0,\infty)$

$\therefore\quad D(\ln(\ln x))=(1,\infty)$

$\therefore\quad D(\ln(\ln(\ln x))=(e,\infty)$

But I wante to find a 'rule' that I could use for any composite function. My internet search left me with the 'rule':

$$D(f \circ g)=\{x | x \in D(g) \cap g(x) \in D(f)\}$$

So I tried my hand at this 'rule' on the question and got

$D(\ln x)=\{x|x \in (-\infty,\infty) \cap x \in (0,\infty)\}=(0,\infty)$

$D(\ln(\ln x))=\{x|x \in (0,\infty) \cap \ln x \in (0,\infty)\}$

$\qquad\qquad\quad =\{x|x \in (0,\infty) \cap x \in (1,\infty)\}=(1,\infty)$

$D(\ln(\ln(\ln x)))=\{x|x \in (1,\infty) \cap \ln(\ln x) \in (0,\infty)\}$

$\qquad\qquad\qquad \,\ =\{x|x \in (1,\infty) \cap \ln x \in (1,\infty)\}$

$\qquad\qquad\qquad\:\ =\{x|x \in (1,\infty) \cap x \in (e,\infty)\}=(e,\infty)$

Which looks like a rather long-winded way of going about it. Is there anything wrong with my notation? Can you suggest a better method other than inspection?

Answer 1

The short answer: No.

Within certain classes of functions, you may be able to find some ad hoc approaches. In the above situation, you can use the fact that $\ln^{k}(x) \in (a,\infty)$ exactly when $x \in (\exp^{k}(a),\infty)$, where "^$k$" means "compose $k$ times". But in general, the domain varies wildly as you change the functions.