I've been making and teaching the log cabin quilt for decades and have developed a system that I use and teach in classes that has students saying, "for the first time I LIKE log cabin!"
1. I NEVER cut individual logs. I strip piece. My logs are cut as narrow as 1 and 1/8 for personal use, 1&1/4 to 1& 1/2 in classes. I make my hearth block (the center) either the same size and the strips or a slight amount larger. I have also cut the center much larger - 3" - and used that space for particular quilting designs.
2. I square up the block after every round. One round = four logs added. I use, and require in my classes, the Bias Square 8 square ruler from That Patchwork Place. It is the ONLY ruler on the market marked with a solid 1/8" lines at right angles in every inch. This allows you to trim the tiniest amount off each side of the block at each round. I make sure the diagonal line on this ruler runs through opposite corners of the center square when I begin to trim. This trick allows you to maintain the straightness, the square setting, of the finished block. **Because the log cabin is traditionally pieced in concentric circles, it is quite easy to cause the block to skew....just like you have when you are working needlepoint - the piece becomes trapezoidal shaped from the constant direction of the stitches.** When you are able to keep two adjoining sides of a square at right angles, then you will keep the block square.
3. I press every log away from the hearth (center square). This allows me to turn the block over and measure on the back, no matter where I am in its construction, no matter how long ago I stopped working on the blocks, and start right off making more blocks exactly the same size. I don't even have to write down what size I cut the center because I can just measure it easily on the wrong side of the block.
4. I always work at squaring up from the wrong side of the block. This allows me to keep the center and logs at the same place on the square up ruler, making all the blocks the same size.
5. When/if the last round creates a block larger than 8" (and it often does), I use one of the models of Olipfa brand 12&1/2" square up rulers that has the same 1/8" markings in two directions that the Bias Square 8 has in the 8" ruler.
I travel to teach workshops, you know!
Jan in VA