Currently the search option for the Activity Log only works within the page that is visible at the time you run the search. As a result you may have to run a search on multiple pages of the Activity Log, which takes more time and increases the odds of missing the item you are seeking. It would be helpful if you search across multiple pages of the Activity Log at one time.
Thank you for the idea. Given the low volume of support for this idea, we do not have plans to make updates in this area at this time.