On the behaviour of $\sum\binom{n}{k} (a-1)^k$ where the sum is on $k\le n(1-1/a)$, for some $a>1$

Question

I just came up with the following limit $$ \lim_{n\to \infty} \frac{\displaystyle\log_a\Bigg(\sum_{\substack{k\in \mathbb{N}\\k\leq n~(1-\frac{1}{a})}}\binom{n}{k} (a-1)^k\Bigg)}{n} \qquad \text{ where } (a>1).$$ I am not able to solve it? Any hints?

Update: By Greg Martin comment, the limit hopefully will be $1$: A complete proof will be appreciated.