Having the ability to review logs in a web-based interface is useful for administrators of a tool so we can diagnose and remedy some problems without needing to reach out to support. The downside is the log needs to be filterable or multiple log files need to exist.
An example of a log that would be useful would be an email log. In the event an email fails to send, I'd like to know why without needing to download the entire log in a zip, extract the zip locally, and browse through it when I am looking for a specific issue (email) and I should know a rough estimate as to when it started failing.
Email is my example in this case simply because that is the error I hit this morning - an email that I expected to be sent at 12:15 AM didn't arrive. IT was doing work on the email server around that time, so I expect that it failed because the mail server was down during the time it tried to send it.

One HUGE advantage to this was noticed in a different tool called Jira Software. The tool has a log analyzer that scans through the log for errors and then presents the administrator with a list of errors, written out in plain English so it is easy to understand, and includes a link to the Atlassian support article on how to correct the error. It isn't 100% accurate, but it is much nicer than spending 2-3 days working with support to resolve an issue that I can fix myself by looking at the log analyzer.