I am reading a proof that the Subset Sum decision problem is NP-complete.
I know that the time complexity is always calculated based on the number of bits of the input in binary, hence the $\log{W}$. Also, for this specific problem the size/length of the input is $(n+1)\log{W}$, with $n$ and $W$ being two integer variables ($n$ is the number of available elements in the set and $W$ is the sum we are seeking).
However I am having some trouble when trying to understand the terminology be linear/polynomial in the input size, specifically with these two lines:
The input length is (n + 1) log W, and the running time of O(nW) is not polynomial in this input length.
and
takes O(n log W) time, linear in the input size.
How can I mathematically prove the above two lines? Or what is the mathematical reasoning behind those two statements?
I don't have this conceptual problem when there is only one variable. For example, let's say I have an algorithm with a complexity of $O(x)$ and an input size of $\log_2{x}$. Then, I can express $x$ in terms of the problem size like this: $O(2^{size})$. Now, I can see that the complexity is exponential in the size (in bits) of the problem.
But I don't know how to do this when there are two or more variables in the complexity function (in this case $W$ and $n$).