Do you have to have Logos for computer in order to purchase content and use it on mobile?
No you do not.
Feel free to browse the Logos.com website. When you see a book you want, please ensure that it will show on mobile. It shows just above your price.
Welcome! [:)]
TC is right. You don't have to have a computer. What you can't do, which may be the reason you asked, is make "in app" purchases. Apple charges a 30% fee for all "in app" purchases, which is why you can't do so in apps like Logos or Kindle.