Intuition for $e$ and natural logs

Question

I understand that $e$ is "the growth constant", and I understand that it was derived by: $$ \lim_{ n \to \infty} (1 + 1/n)^n. $$ But how does raising $e$ to $rt$ work, wouldn't it be like raising $$ \lim_{ n \to \infty} (1 + 1/n)^{nrt}? $$ Why does that work (if it works at all), and is there any intuition behind how. Also why is this useful? Thanks in advance!