Well since nobody else has answered...here is my 2 cents. A jack or any kind of brace. Fortunately log homes have overhanging log ends. So a house jack, a big hydraulic jack can be used to pick up the load while work is being done on logs below. Unless you are working when there is a bunch of snow on the roof (highly unlikely), there are probably only a few tons of weight, so as Alan D. suggested to me a hi-lift jack would do the trick. or several of them. Good Luck!
Ok, my ears are burning...been a while since DIY Zack posted this, but I'll add another $0.02 as well. Please take what I say with a grain of salt, as I don't know what I'm doing for the most part and just a DIY person also.
I didn't come up with the idea to use a hi-lift jack myself, in fact where I learned it from is from McRaven's book, that is how he shows to replace a log on a traditional appalachian style home. But let's look at the above pic to understand some of the specifics and some of the work that needs attention.
As you have probably figured out, you need to lift the upper section above the rot. In the pic above, that means that you will only be lifting a minority of the walls which form the corners and only above the rot. Another type jack that would work also is what is referred to as a railroad jack, one of the more popular ones is made by a company, Simplex. These have been around for many years and often used on the corners to raise an entire home. The one I own is rated for 10,000 lbs., and I use it mostly to lift heavy machinery, like my metal lathe which weighs 4,000 lbs. This is what the jack looks like:
The problem with the above style of jack is that the toe is pretty narrow, so in order to use it you need to either extend a timberr under the portion wanting to lift, and grab it with the toe, or get the entire jack underneath so that you can lift it safely with the top portion.
The reason a hi-lift works for this is that the toe portion of a hi-lift extends longer, making it easier to grab under the log which needs to be lifted.
For ease of discussion, I'm posting a pic of a hi-lift, which are available much cheaper than a railroad jack BTW.
The Hi-Lift pictured above, which is available from Northern Tool is rated for 7,000 lbs. Not bad for $90. The Simplex railroad style has a list price of over $1k. The other thing is that any of these jacks will scale the load, IOW, if you use 2 hi-lifts on the same corner you could have 14,000 lbs. lifting. You still need to get the toe under the log and/or figure a way to secure it, but you would need to do that with the railroad jack anyway.
Also, you should consider that you will be lifting a small amount of the full home in this case, as the rot goes up so high. That home is in serious needs of repair, hopefully that amount of work will not shy you away as you find out how much rot is there, it is SUBSTANTIAL.
Looking at the picture you posted, the bottom two logs on the left side look solid, but the 3rd one up (under the window) has some rot on the top, you should consider replacing at least as much of that log that has rot, which probably extends at least to the other end of the window...that much is not rocket science as the rot comes from the water that leaks in and/or around the window openings. The other side of the corner needs to be replaced up to the top of the photo, but not clear how far along that wall rot exists. It is SUBSTANTIAL as I mentioned.
The way I would approach it is to first remove the window so that you can save it if possible, otherwise you will need a new window, and that might be advisable anyway and you will need to decide that. I would try to lift both sides of the corner (i.e., 2 hi-lifts) at the log above the window to allow you to cut out and replace the lower logs underneath. It may require some type of fixturing to get something under that log, but the hi-lift toe could do it by itself possibly. The rest of cutting out the rotted sections, replacing the logs, and lowering the top portion back down after it is all cleaned up.
That is a major project, and I'm not trying to discourage you but it will take some time. Given some patience and tenacity, it is doable by a DIY.
That said, you should get some advice from someone that repairs log homes professionally. As I mentioned, I'm just a DIY type person and there have been people point out on this site how much I DON'T know. I am warning you up front.
I won't give you a lecture on getting a log home inspector to look at such a home before you purchase it, but suffice to say it would have been a good idea. I would be curious how you have repaired this, or if you have attempted to after you posted almost 2 months ago.