Log homes can be built upon a fairly standard continuous foundation.  John put his on continuous foundation with an inverted-T footer, and it has a crawlspace. 

Log homes can actually be built on just about any kind of foundation that you can imagine: pier blocks, full basement, cinder block, insulated concrete panels (ICFs), slab, et cetera.

While a log home can go on just about any style of foundation, you have to do a few things differently than you would for a regular stick frame home.  That's because a log home is much heavier, and tends to use support logs in a rather unique manner.