Couple ways to do this.
====== script command ======
<code>
mkdir ~/Documents/Logs
sudo nano /usr/bin/terminalLogging
sudo chmod 755 /usr/bin/terminalLogging
</code>

Put this into the file:
<code>
#!/bin/bash
/usr/bin/script ~/Documents/Logs/`date +%m.%d.%y-%H.%M.%S`.$$
</code>

===== Integrate with mac terminal =====
In the terminal options, set to run:
<code>
sh terminalLogging
</code>

===== About =====
<code>
man script
</code>

====== iTerm ======
Get http://iterm.sourceforge.net/

Shell > Log > Start / Stop