Hi,
I have had 5 shiitake logs under my front bushes for years. While I thought they had finally pooped out, they produced one flush earlier this spring. I have had these logs for maybe 13 years, when they were expected to have a useful life of 5.
I took care of my logs to make sure they were in shade, otherwise I let nature take its' course. I have had friends who watered them regularly like they did their garden and had regular flushes. I have a friend who has an indoor pool whose shiitake logs are continuously fruiting. However, she thought they tasted like pool chemicals!
The Shiitake Growers Association has a shiitake log innoculating class around the beginning of March each year in Wisconsin. I have never gone, though I am considering going next time. I believe for around $10 you get the log, supplies and instruction plus a lunch ... so maybe it is more than $10.
Below is a picture of a shiitake log from the North Park Village Nature Center at Pulaski and Peterson:
There are all sorts of tricks to get these logs fruiting aggressively. Like soaking them in tubs, which makes sense. Another is boucing the log off the sidewalk, which doesn't make sense but many swear by it.
Have fun!