If you lose 5 pounds in 30 days then you are burning roughly 583 calories more a day than you are eating. You say you eat around 1300 calories a day and no more than 1500. Does that mean you average 1300 calories a day, including occasional higher calorie days. Or, are you saying that you eat 1300 calories a day and occasionally eat 1500. Let's say you eat 1300 a day for 5 days of the week and eat 1500 calories on 2 days. Your average would be 1357. So, do check that.
Let's say your average is 1300 though. Then if you lost 5 pounds in 30 days you would be burning an average of 1883 a day.
I don't know how old you are. I looked at this calculator to get your BMR. If you were 20 it would be about 1838. If you were 60, it would be about 1650.
Of course most people burn more than there BMR. I work from home and unless I do formal exercise I can end up burning only a couple of hundred calories more than my BMR. That calculator says my BMR is 1439. Last month I was fairly sedentary (knee injury) and Fitbit said I burned an average of 1656 a day. In January I did a lot of walking (although usually not for more than an hour at a time) and it says I burned 1786 a day.
So I would guess that most people burn at least a couple of hundred calories a day more than their BMR depending on how active they are.
So, you might look at that to see how well your losing matches up with your calories burned.