How to tackle Big O proofs that involve multiple parameters
That isn't true. As you say, it implies there are $c_2$ and $N$ so that for all $n \ge N$ you have $a^n \le c_2 \log n$. Consider:

$\begin{align*}
  \lim_{n \to \infty} \frac{a^n}{\log n}
      &amp;= \lim_{n \to \infty}
                \frac{a^n \log a}{1 / n}
      = \infty
\end{align*}$

Here we used l'Hôpital. But that the limit is infinite means that eventually the ratio is larger than any given $c_2$. In fact, it proves $a^n = \Omega(\log n)$.