I'm no IT guy so let me ask what are the options for monitoring the IP's looking to get access to your network? If I have a cable modem that's connected to a wifi router which also has a LAN hub and I don't actually have a PC running at all times to do that kind of monitoring -- does the router log this stuff so a PC on the network can periodically look at the log?
For the vast majority of consumer level kit, no.
I have a, now relatively ancient, professional Cisco router that is also configured as a firewall. This is configured to log, via syslog, over the network to a server that runs 24/7. That server has two network cards and is also configured as a firewall (linux, ipfilter) which also does its own logging. Furthermore the server regularly polls the router, via SNMP, to get more statistical information from the router that's also stored and graphed on the server. It's all a bit of a pain in the butt to configure but that kind of thing used to be my day job.
Some consumer level kit is better than others and, with the right software, will provide quite useful monitoring; but most consumer level kit is, from this perspective, quite useless.
Whatever you're using, if you want monitoring and logging at the level that would record individual IPs for firewall logs you're going to need something turned on 24/7 to store the logs. Kind of obviously, if you're going this for security monitoring you want the device recording the logs to not be the same device recording the logs, so that if the latter fails or is compromised you still have the records stored on a working, uncompromised device.
Advice on how to do this is going to be, at least in part, dependent on the specific kit you're using and is often best found on forums dedicated to that kit. For consumer kit, your best bet is using user supported replacement firmware for the platform - things like OpenWRT.