TaskLogs option outputs in the XML the log of all the scheduled tasks (workflow execution and publishing) that have been executed for the object.
For example, if the 'video' object with id 41314 has been scheduled published to '/Default' and if an attempt to generate its video preview as a scheduled task failed, executing: ../do.ashx?cmd=objsnav&ids=41314&options=tasklogs&max=5 produces:
<video id="obj41314" . . . >
. . .
<nav:taskLogs handle="1">
<nav:taskLog
id="17450" time="2011-11-01T20:20:21.327Z" result="Success" type="Publish" priority="0"
execProc="\\MIMO-PC64\Tera Back4" profile="" objectid="1595" cmd="aids:41314,41274;" userId="4">
<message>
Completed task with id 909
Successfully published on destination '/Default' (id 1595)
Published 2 video ('11902', 'VIDEO_006').
</message>
<user idref="obj4">
<nav:refObject idref="obj4" />
</user>
<object idref="obj1595">
<nav:refObject idref="obj1595" />
</object>
</nav:taskLog>
<nav:taskLog
id="17449" time="2011-11-01T20:20:06.317Z" result="Error" type="ObjsWF" priority="0"
execProc="\\MIMO-PC64\Tera Back4" profile="" objectid="5783" ids="41314 41274" userId="4">
<message>Error in the activity 'videoConvertActivity': 'Error calling the plug-in 'ParseMultimedia.dll': "The system cannot find 'FFMpeg.exe'" (ERR0503)' (ERR1011)</message>
<user idref="obj4">
<nav:refObject idref="obj4" />
</user>
<object idref="obj5783">
<nav:refObject idref="obj5783" />
</object>
</nav:taskLog>
</nav:taskLogs>
. . .
</video>
The logs are listed in decreasing order of execution time, so the most recent one is at the top.
As for the 'Tasks' option note how the user the complete XML of the user that scheduled the task and of the object associated with the task are expended within each task ('user' and 'object' element respectively).
See also