You can absloutely make a log about your G1/P1. In fact, please do. They are less complicated than the new ones, so there will be less to write, but it will be a great reference for those who don't own one or are considering purchasing one. I used to log mine all the time back when they first came out, because they used to sell official and unofficial Tamagotchi log books in stores next to Tamagotchi.
At the very least, you should log end-of-day stats, and any major events such as illness, evolutions, running out of hearts, etc. I even used to log what times during the day I fed/played with/cleaned my G1's & G2's. It definitely helped me to know all aspects of my Tama inside and out.