script command

mkdir ~/Documents/Logs
sudo nano /usr/bin/terminalLogging
sudo chmod 755 /usr/bin/terminalLogging

Put this into the file:

#!/bin/bash
/usr/bin/script ~/Documents/Logs/`date +%m.%d.%y-%H.%M.%S`.$$

Integrate with mac terminal

In the terminal options, set to run:

sh terminalLogging

man script

Shell > Log > Start / Stop