Google Public DNS stores two sets of logs: temporary and permanent. The temporary logs store the full IP address of the machine you're using. We have to do this so that we can spot potentially bad things like DDoS attacks and so we can fix problems, such as particular domains not showing up for specific users.
We delete these temporary logs within 24 to 48 hours.
In the permanent logs, we don't keep personally identifiable information or IP information. We do keep some location information (at the city/metro level) so that we can conduct debugging, analyze abuse phenomena and improve the Google Public DNS prefetching feature. We don't correlate or combine your information from these logs with any other log data that Google might have about your use of other services, such as data from Web Search and data from advertising on the Google content network.