I say you can.
Are you with me?
Step 1: Tools of Your Trade
Step 2: Cut Down the Tree
I'm not going to go into detail with this step. You can pull it down with a truck and chains (i doubt it), you can chop it down with an axe (unlikely) or you can chainsaw that baby to bits (your best bet). If you are fortunate enough to come across a downed tree or a goodly sized chunk of unhewn timber consider yourself advantaged. The size doesn't really matter. It's based on what you want. The piece I'll be using is about 30 inches long by 14 inches wide.
Step 3: Cut Out a Notch
Only cut a small enough section that you can fit your chainsaw in sideways. We're going to be tipping this baby over to save us some time. So, you have two cuts, a third of the way through the log, no farther apart than maybe 8 or 10 inches. Start at the top of one of the cuts and cut diagonally towards the other. This will make it easier to notch out this section. Repeat with the other side. If you get a pesky little piece of wood you can try using the chainsaw or if you have a sledge hammer nearby you can pound it out. If you go the chainsaw route avoid cutting deeper into the log than any other cut.
I cannot repeat this enough. Avoid cutting deeply into the log. It will look stupid and if you want to fix it this will take a lot of time smoothing it out.
Step 4: Cut Out the Rest
Once you're in position cut down the length of the log being sure to not veer more than half to a third deep. Keep cutting until you've got about 8 to 10 inches left. You'll want a chunck of wood on the end to ensure this looks like a log and not a board. Unless of course you want a board looking sign, then just keep cutting. In fact, you should just the plane the whole thing. Or don't read any of this b/c notching out this piece is the hard part. Etching is easy. Just skip to the end if you want a board with an etched in sign.
In short, cut from A to C. Then from B to C as shown in the picture.
Step 5: 4a - My Mistake
For your sakes, i'll post what to do. It will be painful, but it is possible to salvage that piece of junk you just made. Here's what to do: the image shows a bunch of cuts that are not to the same depth. Each green line is a cut with you chainsaw. They are not straight nor are they to the same depth. The red line shows the unevenness of the writing surface. Ignore the 'This Piece Is Now Gone' label as you skipped that part. You will need a chisel to smooth out the really rough edges. Most of the edges will be really rough. This will take a lot of work and sweat, but you can dramatically improve the appearance of your writing surface.
Once you've chiseled your knuckles raw grab the belt sander. No sanding disc or orbital sander will do unless you opted for a relatively small log to begin with. Sanding will help smooth out the initial rough spots and the improved rough spots you made with your chisel and now cramping hands.
It will not look perfect.
I warned you.
But it will look good and I think you'll like what you end up with. How do I know this? I made this terrible terrible mistake and had to use a chisel and belt sander too. My finished product looked pretty good. At least for me. Something I can plop in the yard and not worry about seeing some of those deep gashes from a distance.
Step 6: Smoothing
Tip:we'll be applying an optional polyurethane finish to protect the sign outdoors. A rougher surface will absorb more polyurethane which means you'll need to apply more of it. The smoother you get this the better, but remember it's going to be outdoors so it doesn't need to look nice and smooth. It just has to be smooth enough to accept the polyurethane easily.
Step 7: Write It Down
Something to consider is how many letters you have. The fewer you have the easy it will be to eyeball their spacing. If you have more you would probably benefit from a little measuring.
Step 8: Etch Your Letters
I've heard you can use small amounts of acetone/nail polish to paint on your letters then light them on fire then snuff out the fire once the burn effect is reached to your liking. I haven't done this, but if it works it may be faster than etching. Any feedback on this would be great if you've done it. Please be careful, you're using fire. You should maybe think about doing this outside.
Step 9: Last Step-Preserve Your Sign-Optional
A thick coat or two of outdoor polyurethane will ensure your sign stands the test of time. Remember when I said a smoother writing surface will decrease the amount of polyurethane you'll need? You'll really notice this principal when applying it to the bark. You can skimp on the bark and focus on your letters if you like.
When you're done let it dry for a day or two. The final product will look something like this. I hope you like the results of your hard work.