Ok, I guess first thing to check is connectivity. Write down a couple of ip addresses and keep them handy for the next failure.
If you can't get out at all then you need to make sure this isn't Base-station problem, you can just plug directly into the cable modem for a few days and see if this isn't just a cable modem problem. Some cable modems have management ip's and you just type into your web brower an address like 192.168.100.1(this address may not work for your modem) and get the status of the modem. If would also be a good idea to call the cable company and have them check the levels on the cable modem.
If it isn't a connection problem go into terminal.app you can find if you go to the Applications folder and then utilities folder.
Type into the terminal window.
cat /etc/resolv.conf
You should see two lines starting with the word "nameserver"
Looking at the logs you sent they should be 24.217.0.5 and 24.217.201.67.
You should be able to then be able to use telnet to test the connection to the DNS servers type in what ever address is in the resolv.conf followed by 53
telnet 24.217.0.5 53
and get something like this... Try both address
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207-218-58-55:~ lostngone$ telnet 24.217.0.5 53
Trying 24.217.0.5...
Connected to vip1olvemo.stls.mo.charter.com.
Escape character is '^]'.
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You can also try the dig command during a failure and see if you get an output that looks like the following...
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207-218-58-55:~ lostngone$ dig
; <<>> DiG 9.4.2-P2 <<>>
;; global options: printcmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 64387
;; flags: qr rd ra; QUERY: 1, ANSWER: 13, AUTHORITY: 0, ADDITIONAL: 0
;; QUESTION SECTION:
;. IN NS
;; ANSWER SECTION:
. 349001 IN NS g.root-servers.net.
. 349001 IN NS k.root-servers.net.
. 349001 IN NS j.root-servers.net.
. 349001 IN NS l.root-servers.net.
. 349001 IN NS h.root-servers.net.
. 349001 IN NS m.root-servers.net.
. 349001 IN NS i.root-servers.net.
. 349001 IN NS f.root-servers.net.
. 349001 IN NS c.root-servers.net.
. 349001 IN NS e.root-servers.net.
. 349001 IN NS a.root-servers.net.
. 349001 IN NS d.root-servers.net.
. 349001 IN NS b.root-servers.net.
;; Query time: 10 msec
;; SERVER: 209.165.131.12#53(209.165.131.12)
;; WHEN: Sun Jan 4 13:47:56 2009
;; MSG SIZE rcvd: 228
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I have had a problem like yours but it turn out to be my ISP's crappy DNS servers. It took 3 weeks of calling then to finally to get them to admit they were having name resolution problems. I used these guys until they got the problem fixed.
http://www.opendns.com/