Need a Eurail pass?

I have never purchased a Eurail pass before. I am traveling from Barcelona to Logrono, Spain.  I want to take the train at 8:45am and it arrives in Logrono at 12:52pm.  That’s the only train time that works for me.  On the return one week later, I need to go from Logrono, Spain to Barcelona airport.  Again, only one time that works, the 7:30am train, arriving at 1pm to airport.  I can see the different train companies online to buy a ticket directly from the individual train company.  Do I also need to get a Eurail pass?  I’m not sure what I will be doing regarding traveling within the week I’m there, but I expect I will be staying in the Logrono area the who time. Thank you for your help in explaining what I need to purchase.