Say size absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer n $n$ that is reasonably large, can we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ is very close to $n$ say within $O(log(n))$? If I cannot find for all O(log(n))$ since there are only $n$, then what log^{O(k)}{n}$ such $a,b,c$ combinations(answer from Petrov).

What is the lower bound on $n$ as a function of $k$ upto which I can find such $a,b,c$?

Say size of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer n that is reasonably large, can we always find integers $a,b,c$ such that $|a{b^{c}} - n|$ is very close to $n$ say within $O(log(n))$? If I cannot find for all $n$, then what is the lower bound on $n$ as a function of $k$ upto which I can find such $a,b,c$?