It logs only HTTP traffic. All search engines now use HTTPS.
I monitored my desktop's traffic with ntopng and saw that all HTTPS traffic doesn't appear in Tomato logs.
ntopng is a web-based network traffic monitoring application released under GPLv3. It is the new incarnation of the original ntop written in 1998, and now revamped in terms of performance, usability, and features.
I use for example a Twitter client which constantly streams via HTTPS. In Tomato logs no connection to Twitter visible.