User blog comment:D57799/Some discoveries and guesses about uparrow notation/@comment-1605058-20150128140139

1. In multiplication, this is simply associativity.

3. How do you define $$a\uparrow^3 n$$ for non-integer $$a$$? Already $$a\uparrow^3 2$$ is problematic.

4. To my knowledge, super-logarithm (if this is what you mean) is undefined for non-integers.

6. Why is graph of a^^^x supposed to pass though (-2,-1)?