At least in Ubuntu systems there is a file /etc/apt/apt.conf.d/15update-stamp containing:
APT::Update::Post-Invoke-Success {"touch /var/lib/apt/periodic/update-success-stamp 2>/dev/null || true";};
So see if you have /var/lib/apt/periodic/update-success-stamp and if you have it, you can use
stat -c %y /var/lib/apt/periodic/update-success-stamp
command to get the the time of the last "apt-get update" invocation.
And if your system does not have that apt configuration file, you can always add it.
This answer is based on reading the following files:
/etc/cron.daily/apt
/etc/apt/apt.conf.d/10periodic
/etc/apt/apt.conf.d/50unattended-upgrades
/usr/share/doc/unattended-upgrades/README
which are part of packages (including unattended-upgrades and update-notifier-common) included in the standard Ubuntu installation. There are other tools available from the distribution that could achieve the same ends, such as apticron and cron-apt.
In the file /etc/apt/apt.conf.d/10periodic, add a line
APT::Periodic::Unattended-Upgrade "1";
This configures Apt's periodic scripts to run an unattended upgrade every day. For a longer interval, increase the 1 to a larger number. A 0 for this variable would mean 'disabled'.
In tandem with this, the autoclean line should be enabled by replacing a 0 there:
APT::Periodic::AutocleanInterval "1";
By default, Ubuntu 8.04 only configures automatic unattended upgrades
from the hardy-security archive. This will cover you against security vulnerabilities, but not for general bug-fix releases. For that, you need to enable the hardy-updates archive in /etc/apt/apt.conf.d/50unattended-upgrades:
// Uncomment the line that reads as follows:
"Ubuntu hardy-updates";
Ubuntu's spec for this feature appears on a wiki page.
Note that Debian systems seem to have the same exact infrastructure, apparently from this feature migrating upstream.
Best Answer
The log file you are searching is: /var/log/apt/history.log
Here are 2 log entries: