Ok, so based on your answer that you can't change the app to log with a full timestamp, you still have an option, though it's imperfect.
In your props.conf file, you can use something like this:
[nameofyoursourcetypehere]
DATETIME_CONFIG = CURRENT
This will cause Splunk to "assign" a timestamp of the current time (i.e. the time the file was indexed) to the events. Very likely, all of the events would have the same, or almost the same timestamp assigned since it would only take a few seconds to index the file. But, frankly, without a timestamp in the log, you don't actually know when things happened anyway.
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