It is simple to decide powers of 2 in $O(n)$ time because it's just "0-bit Unary" after bit-1. (eg. $1000$ is a power of 2 in binary).
I haven't found many other trivial powers of $K$ that can be decided in polynomial-time with the binary-length of the input.
Can we decide if a number is a power of any given $K$ in polynomial-time and in a practical amount of time?
Something not naive such as keep dividing $N$ by $K$ until you reach the smallest value $2$ for deciding a power of 2.