I found a solution to the problme of waking up my mac that does NOT require a second computer on the same LAN. I only have one computer at home always connected (wired) to my home network. If the computer is sleeping when I use logmein to access it, I would only get a black screen and could not get the computer to wake up. I have discovered if I try to use logmein to access my home computer from both a remote computer and my iPhone (using the Logmein iPhone app) at the same time, my home computer wakes up. Whichever device logged in first disconnects and the second one can access the computer normally. Hope this helps others who have encountered this problem.
