Here are some or my logo designs from scratch using either raster-based iPad apps and stylus in hand like Procreate or Illustrator and the pen tool. The raster work has a different and more hand-drawn effect, but they do not scale so well as the vector-based Illustrator images (all are presented here below as raster but some were only exported after scaling at the end as raster).
Thus, for websites and logos online, raster logos or ones made with apps, such as Photoshop or Procreate, are just fine. But if you need to print them as business cards, use Illustrator and save as PDF. I can do both as you can see below:
Here are some logos done for individual clients blown up a bit and rescaled individually (Informance was not submitted in time so it was unused by the company and InvoiceOcean was a company I worked for and used these logos or images below in the Adwords visual campaigns):