An inventor's log book is an extremely important document! Mind you, so is a log book for many other purposes as well.
When a patent is applied for there is a need to establish "prior art" - the knowledge on the particular subject which came before the application. Generally this will be in the form of other patents, or, published material. But the exact date of your own work needs to be established as well.
Everything you do, however inconsequential you fell it might be, towards your invention should be written down. You should have someone witness your work (signing on each page with you) and date it. Each page should be numbered sequentially from start to finish and pages must not be removable. Standard exercise books, loose-leaf books and the likes are not acceptable - they can be tampered with. Sewn or stitched bindings are to be preferred.
Some stationery suppliers produce special books for just this purpose, copy the pages using a light copy setting and the background grid or lines disappears. But copy it on a darker setting and they have versions that will add "Do Not Reproduce" to the page for added security. Sewn ledger books are fine, but make sure they have white pages, the blue of some ledgers will not reproduce well.
Write neatly, by hand (obviously) with preferably a black pen. Uniball roller pens in black work well for me. Make clear sketches of your drawings in the log book, even if you are creating detailed drawings on a CAD package. Be detailed, without going over the top. Remember it is likely that if ever your log books are needed they will be read by expert witnesses, attorneys, judges and other inventors. Never add anything to the book, don't tape or staple any additional pages into it. If you have drawings that will not fit, then sketch the basis of the drawing on the page where it will be dated and witnessed. Then create the drawing with a reference number that is in your notes, and then have your witness sign and date it as well. It should be obvious to anyone looking at the log book and the drawing that the sketch in the log book is in fact related to the drawing. If you update a drawing, get it witnessed and dated again and refer to it in your log book as before.
Inside the front cover of the book clearly write your name, the date the book was started, and when finished, the date it was finished. Another suggestion is to number your books in sequence. Store your books securely, they can be very valuable in the case of a dispute.