1. 2^3 = 8
3^2 = 9
so 2^3 < 3^2
2. 2^0 > 0^2 1>0
2^1 > 1^2 2>1
2^2 = 2^2 4=4
2^3 < 3^2 8<9
2^4 = 4^2 16=16
when x < 2, 2^x > x^2.
when x=2, 2^x=x^2.
When x > 2, 2^x < x^2.
However when x=4, we get 2^x=x^2, then when x>4, 2^x>x^2
This is as far as I got with 2.
3. I noticed that,
2^4=4^2
3^6=9^3 so: a^2a=(a^2)^a
6^12=36^6
By comparing, a^2a=(a^2)^a with a^x, x^a
We know a^2 must represent x so a^2 = x
However, in the first equation, we have a^2a, not a^(a^2)
in fact, a^2a < a^(a^2)
a^2a < a^x
so, a^x must be greater than x^a
3^4 > 4^3 81>64
5^7 > 7^5 78125 > 16807
3^20 > 20^3 3486784401 > 8000
9^8 < 8^9 43046721 < 134217728
15^7 < 7^15 170859 < 4.7x10^12
so, when a<x, a^x > x^a