There is no way currently to export a log to the cloud from within the script. This makes the export not all that useful since it cannot be automated from within the TestComplete framework making the ability to use it very limited.
Should be able to export the log in similar manner to what we have for the local log files. Using the SaveResultsAs function of the Log object to create a shared log on the Cloud instead of a local copy. This would obviously need to return the location of the log on the Cloud. Should take parameters to support the same options that appear in the Share Selected Log dialog currently(name, storage period).