does mac keep a log?

Possibly, depending on how you installed it.
If you go to /Application/Utilities/ and pick Installer, there is a log of what has been installed through the installer package system. (File -> Show Log)
Apps that have their own installers might leave messages in the console.
Apps that you simply drag from a folder or disk image and into the apps directory prob don't get logged, though they will more than likely have preference files created in ~/Library/Preferences.
 
Some apps also leave a receipt behind... but I am not sure if all do. I guess it's only true for the programs that use the installer program from apple.
 
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