The Chrome browser allows users to save console logs to a file. One way to achieve this is by using the Jam extension, which provides a single click solution to share all console logs and network requests with developers. Alternatively, users can manually follow four steps to save console logs: opening Developer Tools, switching to the Console tab, locating the logs area, and saving it as a file. Additionally, users have the option to use the Jam extension to automatically save their console logs for them.