In the upper-right corner of the console window, there's two buttons for opening up the Editor and Player logs.

If Unity crashes on you and you're interested in getting the log before re-launching it, you can also grab the log files here:

Editor Logs:

• Windows XP:
  • C:\Documents and Settings\YOUR_USERNAME\Local Settings\Application Data\Unity\Editor.
• Windows Vista, 7:
  • C:\Users\YOUR_USERNAME\AppData\Local\Unity\Editor
• OS X:
  • ~/Library/Logs/Unity/

WebPlayer Logs:

• Windows XP:
  • C:\Documents and Settings\YOUR_USERNAME\Local Settings\temp\UnityWebPlayer\log
• Windows Vista, 7:
  • C:\Users\YOUR_USERNAME\Local Settings\temp\UnityWebPlayer\log
• OS X:
  • ~/Library/Logs/Unity/

Info available from http://unity3d.com/support/documentation/Manual/Web%20Player%20Debugging.html

What I usually do on OS X is view the log file with Console.app (/Applications/Utilities/Console.app) for a live update of things. Its also got a nice hierarchy view on its left for quickly switching between monitored logs.

As already noted, you can open the current editor log by clicking the 'player log' or 'editor log' in the upper right corner of the console window.

In addition, as well as "Editor.log", Unity also keeps the editor log from the previous run in the same location as the current editor log file, named "Editor-prev.log".

On Windows, you can find these files in:

C:\Documents and Settings\Ben\Local Settings\Application Data\Unity\Editor

And on a Mac, you can find it here:

/Users/Duck/Library/Logs/Unity/

However, A neat way to deal with the problem of viewing logs to debug a crash is open your .log files with a text editor which detects changes to a file and notifies your or refreshes with the current changes (such as Notepad++ for Win). This way you'll be notified of changes after unity has crashed, and you just have to OK the refresh message in your text app and you get the new log displayed straight away.

You could even tail the log file (tail for mac, and perhaps BareTail for Win) or use an app which detects changes and refreshes the file, so that you get an external scrolling pane of text showing changes to the log file as they happen. You can also do this for the player log, or even the webplayer log files.