I found some instructions whereby you redirect the whole %APPDATA%\.purple directory to the Dropbox folder, but this resulted in continuous syncing as the buddy icons were constantly refreshing.
So, how did I fix it?
Easy, I set up a symbolic link from the logs directory to point at another folder in the Dropbox folder.
First open up a command prompt, running as an Administrator:
Click on Start, find the Command Prompt icon, right click on it and select "Run as Administrator"
In the window that opens, type the following to copy the old log files to your Dropbox folder:
xcopy C:\Users\<username>\AppData\Roaming\.purple\logs C:\Users\<username>\Dropbox\Application_Sync\Pidgin\adam /s /r
Then delete your original log file folder
rmdir C:\Users\<username>\AppData\Roaming\.purple\logs
/s /q
Then we create a symbolic link from the original log file location to the Dropbox location
mklink /d %APPDATA%\.purple\logs C:\Users\<username>\Dropbox\Application_Sync\Pidgin\logs
Do this on both systems, fire up pidgin and you are good to go!