how do i update an old apple id for a purchased song?

It is bit like me setting up a bank account, but if instead of using my name I can set it up with my only street address as the account owner name. I set it up as 678OakStreet and deposit my money. I later go into the bank and say, "I have moved and I now want the owner of my account to be 324ElmStreet." Or I could go into the bank and say, "I used to be called 678OakStreet, I now want you to set up an account for 324ElmStreet." See the difference? In the first case it is the same account and my money will be there. In the second case I have set up a completely different account which I am doing at the same time as I changed where I live. Now the bank analogy falls apart because unlike transferring money from account to account, you cannot transfer iTunes Store purchases between accounts. If you have something purchased under a different account you can only use it by authorizing the computer to that account even if it is with an old address. If you have not set up a separate account and just changed the identifier (email address) on the account then you should be able to use your authorization you have now. As for the ID you see on the track, it's a bit like me having a bank statement on my shelf at home from 10 years ago saying I have $100 in my account at my old street address. Sure, I have moved but that isn't going to change what's printed on the piece of paper. If I go to the bank and get a new statement it will still say I have that $100 in my account but my new statement will have my new address on it. Your music file has the old address embedded in it.

As Niel says, if you changed your email on your account, you're using the same account. If instead of changing it you signed up again with the new address then you have a new account and will have to live with managing purchases across two accounts which is a nuisance.

Was this purchased with an Apple ID for which you then changed the name (email address), or was it purchased with a different account?

Authorization should ideally track Apple ID name changes in one account. Purchases made with different accounts can never be merged into one account and you will need to authorize the computer to the other account.

Authorize your computer in iTunes - https://support.apple.com/HT201251

Are we talking about one account or different accounts? You can have one account that has gone through many AppleID changes, just the way I have had many different street addresses and changed them as I move. So was this purchased with the same account you are using now just with a different ID at the time, or was it purchased with a different account? Either way the answers for the different scenarios have been provided above.

I have never bought anything from the iTunes Store (only free apps) so I am not familiar with the details of this process. I am a bit confused by Niel's statement which on face value implies if you bought this under an old Apple ID and try to redownload it after changing Apple IDs but not accounts you would get charged again. Nobody would ever buy anything from Apple on that basis because people change email addresses all the time.

I think this is where it is critical we be clear on whether or not we are talking about accounts or IDs. Unfortunately Apple tends to be unclear in its naming of a lot of things and that contributes to the confusion.

You need to either log in with that Apple ID or delete and redownload the song; the old Apple ID is stored inside it.

(158217)

If you haven't bought it under the new Apple ID and it's still available in the iTunes Store, you can redownload it but would get charged again.

(158221)

They refer to changing the email address associated with an existing Apple ID as opposed to going through the signup process again.

(158233)