The log file doesn't record actual births, but it does record matings. If you have a nursing mom in your village, the father is the last one who mated with her. You'll see entries in the log file that look like this:

Lala mated with Kali

You have to remember that the information in the log file is replaced each time you launch the game, so it must be examined or preserved (by moving or renaming the file) before you launch the game again. \:\)
_________________________
Barbara
Unicorn
Last Day of Work