Reading log came home in July. 15 min a day, parent signs. He reads. Every night. Same dinosaur book. Thirty nights now. Knows it by heart, turns the page at the right word. Pretty sure he's not reading anymore, he's reciting. Do I keep signing? Or push a new book. He gets upset if I swap it. Teacher hasn't said anything. Yet.
Sign it. Ours was one truck book for most of class 1. Moved on by himself around Diwali. Leave new ones lying around, don't swap.
Noting the 'leave new ones lying around' trick for a few years from now. My mother says I read one Kannada book about a crow until the cover fell off, and I grew up reading everything.
In most schools the log is there to build the habit of sitting with a book every day, and he's doing that. One thing you could try without taking the dinosaur book away: he reads it to you, then you read one page of something new to him. Our daughter's teacher suggested that when she got stuck on one series, and over a few weeks the new book took over.
My son did the same with one Telugu story book in class 1. I signed every day. With his little sister around, 15 quiet minutes was a win for me. After some time he asked for new books himself.
It might help to check what the log is meant to record first. Our elder daughter's school asked for the title of each book, and a repeat was fine as long as it was written down. Another school a friend's child goes to expected one new book a week. If there's a title column, write the same title thirty times and let the teacher say so if she wants something different.
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