Could you please explain why lne^2 = 2, e^lne^2 = e^2, e^lin2 = 2, and lne^e = e ? Thank you.

Answer 1

by definition,

ln(x) is the power of e you need to get x.
Just as log_10(100)=2 because 100=10^2

b^(log_b(N)) = log_b(b^N) = N

ln(e^e) = e*ln(e) = e*1 = 1