I want to make this, but am not sure how to start. If start with a live-edge log, how would I make the length-wise cuts? I have access to a table saw, band saw, jointer, and planer.
Start by attaching (screwing or nailing) the log to a board (or a piece of plywood), so that one edge of the board is overhanging the log and the other edge is short of where you want your cut. You may have to use wedges to make it stable. It may also help to attach another board of the same thickness to the other side of the log.
Put the overhanging edge of the board against your band saw's fence, then position the fence so that the blade lines up with your desired cut location. Proceed to make the cut by guiding the board against the fence. It will help if you're using a resaw blade.
If you're going to do this a lot you should consider making a sled that runs in the bandsaw's miter slot for this purpose.
Alternately, mill one surface flat on the jointer, rest that surface on the bandsaw's table and cut a straight line by eye (snapping a chalk line would help).
For cuts at an angle tilt the saw's table or position the log on the sled at an angle.
If I had to make this I'd start by getting the short, flat edge (top of the piece in the picture) cut or jointed flat. Then I'd scribe my angle lines on the end as a reference and joint one face.
If the workpiece were short enough, I would use my table saw with the short, flat edge on the table to get my angled cuts. This would be the easy, fast route. If it's close, I might even start my cuts this way, then head over to the bandsaw and tilt the table to match the cut and finish the cuts this way before cleaning up the surface with a hand plane.
If I'm working with a larger piece, I would probably start by jointing one face down to the reference line. You can get a bias to one side going on the jointer pretty easily by putting the pressure on that side.
Once I had a single face, I'd use a planer with a sled (use shims/wedges to set the angle and hot glue to attach the work piece) to bring the opposite face down to the reference line.