Logic puzzle

Hi, I'm trying to generalize the answer to this puzzle:

The King has 1000 wines, 1 of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician. The king offers you his expendable servants to help you test which wine is poisoned.The poisoned wine is very potent, so much that one molecule of the wine will cause anyone who drinks it to die. However, it is slow-acting. The nature of the slow-acting poison means that there is only time to test one "drink" per servant. (A drink may be a mixture of any number of wines) (Assume that the King needs to know within an hour, and that any poison in the drink takes an hour to show any symptoms).What is the minimum amount of servants you would need to identify the poisoned wine?

So when we know only one bottle is poisoned we can use base 2 and deduce that the answer is log base 2 of 1000, and we obtain the answer 10 servants are needed. I'm not sure how to generalize it when say K bottles are poisoned however, any hints? I thought about maybe using a hyper-geometric distribution of NcK somewhere, tbh I'm not really sure...