Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

Question

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible because$\ b+c \ln n$ is always transcendental, while$\ b f(n)$ is always rational. I'm trying to prove it, though not having a too good time. I'd like to read some answer.

Answer 1

Clearly $f(n)$ constant does not work. Moreover for any non constant polinomial function it holds $\log n/ f(n)\to 0$, so no polinomial function $f$ satisfies $a^{b+c\log n} \sim a^{cf(n)}$.