First, I suggest the combination of a robust knife plus a military-grade spirally-wound wire saw. The pair is far more versatile than a larger knife alone, and the pair weighs less than a larger knife plus its sheath.
As a sidenote, one can saw part way through a log and then bash the partly sawn log onto another log or a rock to split the log. There are several videos on YouTube showing the technique. After I watched the videos I decided that a light weight, robust, full-tang carbon steel knife plus a military-grade folding wire saw (to make an improvised bow saw for serious fire wood cutting and shelter building) would work for me. This video shows the use of a folding saw to split wood. A lighter folding wire saw can do the same job for less weight.
Here is another YouTube video showing the use of the folding saw blade in what looks to me like a Leatherman multi-tool to cut and the split a branch.
In the 5" to 7" size range, you might start looking at the ESEE RAT 5 knife
You might also consider some of the larger Fallkniven knives:
In that size range you are more into Bush Craft knife and survival knife territory than you are ultra-light backpacking knives. You will find lots of reviews on BladeForums.com and BushCraftUSA.com
YouTube is also a great resource for the larger knives, as people have posted literally hundreds of reviews, field tests, and destruction tests.
You will have to make a decision as to whether you want a high-carbon (prone to rust) blade or a stainless steel blade. There are excellent knives available with either, although a really good stainless knife with edge-holding capabilities similar to a good high-carbon steel blade will (in general) cost more than the high-carbon steel blade.
You will also have to decide whether you will be using the blade in freezing rain, snow, or cold blowing wind. If yes, I encourage you to consider a plastic or rubber full-wrap handle rather than a full-tang knife with scales that leave the tang exposed (and your fingers can freeze to the exposed full tang; I know, I've done it).
For superb stainless knives with rubber (Kranton, or whatever they call it) full wrap handles, the Fallkniven knives are an excellent starting point, IMHO.
In the lower budget range, you might also consider the older Carbon V Cold Steel Master Hunter (on eBay), or the newer laminated San Mai Cold Steel Master Hunter. Both will do the job you describe. Yes, my "other" knife is an old Carbon V (carbon steel) Cold Steel Master Hunter.
Between BladeForums and BushCraftUSA and YouTube I am sure you will find literally dozens of viable candidates for your mission, to fit your taste, hand size, use model, and budget.
You might also investigate the laminated Mora knives in the larger sizes; they are inexpensive, highly robust, and they serve well in the utility role.