The logs regarding apt are in ' /var/log/dpkg.log' and '/var/log/apt/history.log' (and their respective rotated files with extensions '.1', '.1.gz', etc )
First, you have to be sure that you have free disk space on your boot partition (if this partition was configured when installing)
df -H
If the /boot partition doesn't exist or it has free disk space, follow the next step. Else you'll have to remove some old unused kernels in order to have some free space. This can be a tricky process, read some of the tutorials regarding this process (search for 'ubuntu full boot partition' for instance)
The next step is check the broken packages:
dpkg -l | grep -Ev '^(ii|rc)'
Afterward, with this information, you have to study the dependencies between the broken packages in order to proceed to configure them in the correct order one by one (probably you'll have to repeat the process several times on some of the packages)
sudo dpkg --configure package-name
Paste here your broken packages if you need some more help.
Cheers!