Wood contains lignin which is a natural adhesive if you can acheive sufficient heat and compression. Compression causes heat, so all you need is sufficient compression, probably around 20 ton per square inch. That pretty much takes shop presses and log splitters out of the game.
The mold you compress into becomes another problem, just using a pipe leaves the problem of getting the "log" out of the pipe once it is compressed. If you plan to press through the inside of the pipe will need to be polished. An easier way would be to split the pipe into halves and add hinges & clasps along the split to hold the "log" until it has set up.
There are a couple ways of making logs, and both involve moisture. This means you need to hold compression for a period while the log sets up.
If you can't develop and hold pressure the solution is to use adhesives. 2 cheap and natural adhesives are used vegetable oil and cornstarch. Both burn and give off heat in addition to the sawdust. Both also give the benefit of lower required compression to make a "LOG". What pressure you need is determined by too many possibles to predict. With either adhesive, the coating process consumes time. A drum type concrete mixer is good for batching, and you need to determine what amount of adhesive you need according to the quality of the sawdust. Making logs you want minimum adhesive. The process is similar to making jellybeans, or coating seeds.
Even using adhesive there is a cure period during which pressure must be maintained before the "log" can be removed from the mold.
Of course given tremendous amounts of spendable cash, you can move up to continuous extrusion.
