It sounds like the hard part of this process will be translating the logo onto the tree trunk / log.

Drying

As many have said, this is largely a process of patience. If your aesthetics can stomach it, you can affix a metal strap around the top and bottom of the section of the tree trunk, in order to minimize splitting. This is akin to how a barrel has metal bands.

Image Transfer

The easiest way to accomplish this is to

1. print the logo on paper, at the exact size you wish it to be on the tree trunk
2. then place and wrap it where the logo should be, and
3. tape the corners to the tree trunk
4. take your tool (chisel, wood-dremel, etc) and carve through the paper into the tree trunk

This is really all there is to it, but there'sthere are a few techniques that can make it easier

• you can place pins through the paper, into the trunk, near to the pieces that you'll carve away. ThisThis will also allow for the stencil of the paper to remain in place while parts of it are removed in the cutting.
• or, you can use a small bit Elmer's glue to affix all of the paper to the trunk. Then after the cutting is done, sand off the paper with a fine grain sandpaper.

By using pins, you can retain the natural surface of the wood. This is preferable e.g. if you want to maintain the bark (it'll probably rot off over the years, though, unless you lacquer it heavily).

By sanding the glue+paper off, you force yourself to sand all of the tree trunk, or be content with the aesthetics of a sanded area around the logo. You can make the sanding an artful processionalsprocess, too, though.