The idea of whether or not dogs perceive time the way we do has been debated but some people have truly studied the concept and how it differs between aminals and humans. I do believe dogs differ from the way they perceive time. I do not think they know the difference if you are gone one day or 10. I'll give you info from William Roberts, one such researcher to help explain.
His well-accepted ideas say that animals live "in the moment" and have no real episodic memory. They cannot think backwards or forwards -- they are stuck in the present. They do have learned responses but they cannot remember where they came from. This researcher compares it to young children who hae learned how to do a great many things by age of 4, but they normally do not have memories of learning them prior to that time. Animals stay in that sort of phase. They cannot tell how much time has passed. When you come home, they are glad to see you. But if you come home from work today, or come home 3 weeks from now, they still are just glad to see you. Mine get excited when I come back from the mailbox!
Some may say -- but they learn when we come home from work, when the school bus is going to stop in front of their house, when to eat, etc.... He explains all of that as not really a memory of what happened before, but animals using internal biological rhythms and circadian oscillators to determine event prediction. It is really pretty interesting!
You might want to read through what he has to say, I think it will help relieve some of your fears about leaving for a longer time. Here is a short take on his research:
HowStuffWorks "How a Dog's Memory Works"