So a clock can listen to a station WWVB to update time.
My question is:
Could you use the time code to mark your logs in conjunction with a recorder thus doing away with paper logs all together.
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The FCC no longer requires amateur stations to keep a log. So if you wish to reduce paperwork, a trash can is probably the best piece of equipment to employ.
People that keep logs for other reasons (often contesting, or just personal interest) these days often use some software to do it which will automatically populate the time based on the computer's clock. The computer's clock is often set by NTP: network time protocol.
GPS receivers provide another possible source of time, especially useful when the station won't have internet access. GPS receivers are more ubiquitous, cheaper, and more accurate than WWVB receivers. Many people carry one in their pocket in the form of a smartphone, which conveniently can also run logging software.
The times in your log needs to be approximate. having a few minutes of error is not usually issue. Many software logging programs will automatically use the time from your PC.
When logs are used to confirm contacts a few minutes difference is accepted. The exact difference allowed will vary depending on the service being used. I believe Logbook Of the World will accept up to a 10 minute difference and still confirm the contact.
Having said that, some digital modes like FT8 and FT4 require time to be accurate to within a couple of seconds. as these are digital modes that require a PC, it usually more convenient to use software to synchronize to network time servers instead of sources like WWVB.
For portable digital mode some operators use GPS timing signals.