Unless configured to be saved to flash memory, device logs are lost after each reboot.
The mobile connection process, together with other logs, can be viewed via command line with commands logread && logread -f.
If your SIM can send/receive SMS, you can enquire device state by sending a status SMS to the device. You can find more details on SMS commands here.
I would also suggest to configure automatic modem reboot in Services -> Auto reboot section, when data connection fails. That might help to restore connection before the power cycling.
In terms of mobile connection, I would advice to check if the router has a good enough signal quality.
Also, are you using automatically selected APN or a custom value?