On the other hand, 'at last' is used to indicate that you have been waiting a long time for something to happen, as in 'At last -- here's the bus! It's really late!' (This suggests that the speaker has been waiting a long time for the bus to come.) Similarly: 'I tried again and again to pass the exam and at last I succeeded.' (Again, the speaker had to wait before a desired target or outcome was achieved.)
Actually, there are quite a few confusing pairs in English like 'first'/'at first' and 'last'/'at last'. I have a handout about them that I prepared for students in my classes. If you would like a copy of the handout, I would be happy to send it to you as an email attachment. You can reach me at my yahoo dot com address, Rick_Spindel (on this site, I just abbreviate my surname).
Student or Learner