Hello Al Amin,
The answer to your questions is not in an easy or supported way since there is no mention of a log's endpoint in the QEM API section in the user guide.
I did however manage to find some kind of endpoint for logs while checking the network tab in the browser's developer tools, but it seems like it requires the current logs name as a url parameter.
You can see the same if you open up the log view in QEM for a task and also open up the developers' tools network tab.
But this endpoint seems to get the logs contents as strings in an array
So, the short answer seems to be no, i wouldent recommend using it since it is not mentioned.
Maybe you can do something with selenium to have the script automate the click download button for a specific task.