After reinstalling Sierra failed drive???

If your Mac is working as you expect, there is no reason to be concerned about log entries. They normally contain hundreds or thousands of cryptic messages that don't necessarily mean anything is amiss. Log entries with the words "error" or "fail" are only intended to be meaningful for engineers and programmers, and their terminology generally differs from their colloquial interpretations.

The only thing you need to be concerned about is the ability to recover your essential data and files such as photographs, music, or anything else of value that could be lost if your Mac were to suddenly become inoperative, is stolen, or simply misplaced. Your Mac incorporates the ability to back up that information using a feature called Time Machine, and all you need to do is to purchase an inexpensive ($50 or so) external USB hard disk drive. To learn how to use Time Machine please read Use Time Machine to back up or restore your Mac - Apple Support. Connect the hard disk drive and Time Machine does practically everything else, unobtrusively and with no attention from you.

If you need recommendations for hard disk drives that will work with Time Machine, please write back.

Macs are specifically engineered to be used, and not fussed over or worried about. They work best when treated with benign neglect. If your Mac should ever do something unexpected or you think it's not working properly though, by all means post a question here or contact AppleCare first. Never use Google for help, and don't install some magical cure-all product that claims to fix what's wrong. Those actions are responsible for the vast majority of problems reported on this site every day. When in doubt: stop, investigate, and inquire.

Remain vigilant for scams, particularly "phishing" scams that seek to convince you to reveal personal information or to give money to people fraudulently representing themselves as Apple. People succumb to those scams every day, often (but by no means always) "seasoned citizens", and it's heartbreaking to hear when that happens.

The best thing about Time Machine is that you don't have to buy it. It's built into macOS and you already have it. The only item you have to purchase is a place for Time Machine to store its backups (see the * Note below), and the most inexpensive option is a USB external hard disk drive of adequate capacity. In your case a 1 TB hard disk drive would be more than adequate, since you have used only a few percent of that Mac's total available storage.

Some options to consider are:

• Seagate Expansion 1TB Portable External Hard Drive
• Toshiba Canvio Basics 1TB Portable Hard Drive
• Western Digital Elements 1TB Portable External Hard Drive
• Western Digital 1TB Black My Passport Ultra Portable External Hard Drive

None of them offer any particular advantages and I have no particular preference for one over another. They are all are about the same price, $55 or so, from Amazon, B&H Photo, Fry's, NewEgg, even Walmart. They all include the USB cable that connects to the back of your iMac.

1 TB is far more capacity than you are ever likely to need, but smaller capacity hard disk drives don't really cost any less than $50 or so.

When you get the drive, plug it in. Most likely your Mac will tell you "the device you connected is not readable" or something like that, but it will offer to format it using Disk Utility. Once you do that, your Mac will ask if you want to use that drive for Time Machine. Answer yes, and that's pretty much all you have to do until your built-in hard disk drive wears out or fails for some reason – which may never happen anyway.

A corollary of Murphy's Law is that merely having a Time Machine backup seems to prevent the circumstances that require using it. You may never need to use Time Machine, but if you do, $50 for an external drive is the cheapest insurance you'll ever buy for your Mac.

* Note: To be completely accurate, you don't even have to buy anything. You could elect to partition your Mac's internal hard disk and designate the additional partition for Time Machine backups. I don't recommend that though, because a catastrophic drive failure or similar failure affecting the iMac could render both partitions simultaneously unavailable.

** another Note: Apple also sells the AirPort Time Capsule, which is a combination wireless router and networked hard disk drive specifically designed for Time Machine. It's ideal for portable Macs that you might not want to bother connecting to anything. It's also a lot more expensive than a simple USB hard disk drive.

What problems are you experiencing, if any?

The log excerpts you posted are not rendering correctly for me, and I do not know what a "Disc Recording Log" is.

Glad to hear it! Thanks for the update.