John:
If you came to me at the sawmill and asked me to make you a beam that was 8 3/4" x 6" x 15'4" it would take a log that was at least 10 5/8" inside the bark at the narrow end to produce such a beam.
I would probably look for one at least 11" inside the bark narrow end to make sure it would cut out without any wane on the edges.
What I use to determine log size is a chart I created myself. It's a spreadsheet program that takes the size of the timber and generates the diameter needed to mill out the needed timber from a round log.
Basically the old a squared + b squared = c squared system where c is the diameter of the log. a is one leg of the right triangle or one dimension of the beam (8 3/4") b is the other leg/dimension (6") and c is the corner to corner measurement or the hypotenuse of the right triangle, and the diameter of a log necessary to make the beam. To solve the answer you need to re-arrange the formula to find c by saying what is the square root of a squared + b squared? The answer to your size is 10.609” which is just under 10 5/8”.
I hope this helps, good luck with your project, Jim