I can see in your log that the files are uploaded correctly. Based on the time it takes I have no reason to assume there is any reason why this is not the case. I can also see that you have applied remote file quotas, meaning that you are deleting older backups from Amazon S3 automatically as new files are being uploaded. For example, when the 20170524 backup was uploaded the log tells you that the 20170423 backup was removed.
So, your issue seems to come down to one of the following.
Make sure you are looking in the correct bucket and directory. You can verify this information by going into your backup profile's configuration and checking your settings.
Make sure that you are not looking for the backups you have told Akeeba Backup to automatically delete. It looks like you are deleting anything that's over a month old or over 30 entries ago (the actual rule being applied is not logged). If you are looking for one of the already removed backups then yes, they won't be there.
Finally, you are using an antiquated version of Akeeba Backup. The latest version of Akeeba Backup for Joomla! 2.5 and PHP 5.6 is Akeeba Backup 4.7.5. Maybe you'd like to update?
Nicholas K. Dionysopoulos
Lead Developer and Director
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