I'm supposed to be induced at 38 weeks and at the beginning I didn't know my date of my last period can doctors still give me an accurate due date by ultrasound

All I can tell you is that my first child's due date was according to her ultrasound, and she was born on exactly that day. Also my sister just had a baby, and she knew the date of conception, and they changed her due date to be correct according to her ultrasound. I would trust that the doctors know what they are doing. You will be just fine. Congratulations!

It is to my understanding that they can. I was told, that according to my last period, my due date was January 10th, and when they did the ultra sound, it read that I was due January 9th. So even if You have no idea when you last period was, and ultra sound should give you a very good idea of when your due date is.

Actually dating by ultrasound is more accurate than going by your last menstral period (LMP). The reson is because, while you are more fertile around 14 days after the start of your period, that does not mean that is when you conceived. You can conceive at anytime during your 28 day cycle. So you have a wide range of time to work with there. With the ultrasound, they can tell, judging by the measurements of the baby and by what organs are formed, almost exactly how old your baby is, which gives them a very accurate due date. Esp. with the high quality u/s they use today. I would not worry too much about it, you are in good hands.