In increasing order, rank:

Question

3^ln 2, 2^3, 2^ln 3

*do not use a calculator, and explain your choices

Obviously, 2^3=8
So, instinctively, I entered the other values into my calculator just to rank the quantities using decimal places. However, in my calculator, 3^ln 2 and 2^ln 3 both yield: 2.1414886064.
So, can someone explain this problem to me please!

Answer 1

2^3
take ln and get ln(2^3) = 3 ln 2

3^ln 2
take ln and get ln (3^ln 2) = ln 2 ln 3

2^ln 3 also gives ln 3 ln2

and ln 3 <3