How to log login and logout on Mavericks?
Open up a terminal and enter the command: last

chris@chriss-mbp-2:~/Documents/dev/git/$ last

    chris     ttys005                   Sun Dec 22 01:51 - 02:57  (01:06)
    chris     ttys005                   Sun Dec 22 01:50 - 01:51  (00:00)
    chris     ttys004                   Sat Dec 21 13:32 - 19:53 (1+06:20)
    chris     ttys003                   Sat Dec 21 13:32   still logged in
    chris     ttys002                   Sat Dec 21 13:32   still logged in
    chris     ttys001                   Sat Dec 21 13:32   still logged in
    chris     ttys000                   Sat Dec 21 13:32   still logged in
    chris     console                   Sat Dec 21 13:32   still logged in
    reboot    ~                         Sat Dec 21 13:32 
    shutdown  ~                         Sat Dec 21 13:32


Further details about last can be found in the man pages here.