The complexity of $A^{k \cdot log_{x} (n)}$ is exponential?
This is polynomial complexity, because

$$A^{k \log_x n} = A^{k \lg n / \lg x} = (2^{\lg A})^{k \lg n / \lg x} = (2^{\lg n})^{k \lg A / \lg x} = n^{k \lg A / \lg x}$$

and $k \lg A / \lg x$ is a fixed constant.