For fixed $a,b,c \in \mathbb{R}$ with $ac \neq 0$, it seems to me that one can find an increasing sequence of integers $\{\alpha_n\}$ such that the quantity $c \log \alpha_n$ becomes arbitrarily close to elements of the set
$ A = \{ ak+b \,\colon k \in \mathbb{Z} \}. $
First, is this true? If so, my question is: how good is the approximation?
For example, given any $\epsilon > 0$, is it possible to find infinitely-many integers $n$ such that, for some constant $C$,
$ \textrm{dist}(c \log n,A) := \inf_{k \in \mathbb{Z}} |c \log n - ak-b| \leq C n^{-\epsilon}? $
How about
$ \textrm{dist}(c \log n,A) \leq C e^{-n}? $
I am also interested in results which might say something like "There are at most finitely-many $n$ satisfying
$ \textrm{dist}(c \log n,A) \leq C e^{-n^2} $
for any positive constant $C$" to illustrate the "best-possible" nature of a less restrictive bound.
Motivation
I am trying to determine the behavior of a quantity like
$ \left|\left(1-e^{i c \log n}\right)g(n)\right|^{1/n}, $
where $g(n)$ is well-behaved. Until now I have simply been excluding all $n$ for which $1-e^{i c \log n}$ lies in some small fixed neighborhood of the origin, but doing this I lose infinitely-many $n$.
If, for example, it turns out that, for some positive constant $C$, there are only finitely-many $n$ satisfying
$ \left| 1-e^{i(\theta + c \log n)} \right| \leq C n^{-1-\epsilon} \tag{1} $
for any $\epsilon > 0$, then all but finitely-many $n$ satisfy
$ C n^{-1} \leq \left| 1-e^{i(\theta + c \log n)} \right| \leq 2. $
In that case we could exclude at most finitely-many $n$ to obtain the desirable property
$ \left| 1-e^{i(\theta + c \log n)} \right|^{1/n} \to 1 $
as $n \to \infty$.
Now, if we let
$ B = \{2\pi k - \theta \,\colon k \in \mathbb{Z}\}, $
then equation $(1)$ is equivalent to the existence of a positive constant $C_1$ such that only finitely-many $n$ satisfy
$ \textrm{dist}(c \log n,B) \leq C_1 n^{-1-\epsilon} $
for any $\epsilon > 0$.