Can you tell the difference between a battery issue and an alternator issue by symptom?

Over the past three days, my car started to be sluggish when starting, but it did start and worked through the day- yesterday it was driven 10 miles to work, sat for 7 hours and started to be driven back. This morning the engine slowly tried to turn over, but couldn't make it. Yesterday and today the clock showed 12 when the key was turned, so obviously there is a battery issue, but I don't know if I just need to replace the battery or deal with the alternator. No machine gun noises when ignition is engaged, just the battery not turning over the engine. Not quite sure how to handle the problem this morning; would like to get to work!