Within Messaging log you can filter messages so you only see the messages that are relevant for your query.
This can be done in two ways:
When in Messaging log, press the big Filter button.
Campaign: Only show results of a specific campaign
From / Shortcode: Only show results from a specific Sender ID
To / Recipient: Only show messages sent to a specific recipeint
Channel: Only show SMS or Push
Display: Number of messages shown per page
Direction: Only show Outgoing/Message Terminated (MT) or Incoming/Message originated (MO)
Status: Only show messages with a specific status
Message: Only show messages that have certain specific text in its body.
Shortcuts You can also quickly filter on certain values by clicking on a recipient's phone number for example. This will only show the messages to that recipient. This works for the values: