Last Log for Jira comes with a dedicated REST API. The following features can be used through the API:
Available for Jira administrators and system administrators
<base>/rest/lastlog/1.0/log
<base>/rest/lastlog/1.0/log/{filename}
An example link can be found after executing a custom search.
<base>/rest/lastlog/1.0/log/atlassian-jira.log?query=&maxEntries=500&exclude=&reverse=false®ex=false&casesensitive=false
Only available for Jira system administrators.
<base>/rest/lastlog/1.0/threads
Be careful. Killing. the wrong thread can seriously harm your instance!
<base>/rest/lastlog/1.0/threads/kill/{id}