Yes, you can access your Boost Mobile call history via your handset using this linkhttp://m.telstra.com/boost or via #111# selecting option 4 to see the last 5 transactions in the previous 24 hour period. If you require call history further back from this you will need to call the Boost Customer Care team on 125 8881.
Is boost going to add the ability to look beyond 5 transactions? an interactive list of at least the past months transaction would be really helpful! perhaps through an app?