User blog comment:Nayuta Ito/tetrational logarithm/@comment-1605058-20150731105254/@comment-1605058-20150731231610

Let's see if I get what you mean: for any fixed $$n$$, we take $$a_n$$ to be a number such that $$x\uparrow\uparrow n=(y\uparrow)^na_n$$, and then you consider $$\lim\limits_{n\rightarrow\infty}a_n$$? If so, can you show that the limit always exists?