That is... Really not the case, at least for the latest version.
It may have been in the past (and so it may be the correct answer for the OP's request, depending on their version), but currently the log file only contains the output of the console for the latest session.
The saves are located in game/saves. Copying that folder *should* allow you to take your old saves to a new version of the game. This assumes compatibility, however, and that is largely (but not entirely) up to the author: significant changes may break compatibility, regardless of the author's wishes.
Each .save file is a zip archive containing, among other things, a file called "log", which may be how the game used to save the current status when only one save was allowed, assuming that was a thing. In the latest version you get a lot of save slots, however, and each of those is a separate file, so in the future moving your saves will require moving a folder, not a file.