To enable low-level logging in MavensMate, set the following:

"mm_log_level" : "DEBUG"

To specify a logging directory, enter the path to it with the following (must be an existing path):

OSX, Linux users:

"mm_log_location" : "/path/to/your/logs"

Windows users:

"mm_log_location" : "\\path\\to\\your\\logs"