Justine,
A logo ought to (be reworked to) show the actual/original artwork accurately as Barbara mentions, and it can contain clean and simple shapes.
There are quite a few different possible suitable options depending on the actual/original artwork represented by the JPEG.
Whichever you choose, you can lock the JPEG and work on top of it.
You can start you considerations by looking at certain parts under Add and edit content > Drawing here,
https://helpx.adobe.com/illustrator/user-guide.html
including the Draw with the Pen ..., Draw simple paths and shapes, Draw ... grids, and maybe others,
and also taking Drawing basics, Edit paths, and maybe others,
into account.
If you post the JPEG or something like your logo, helpers can make more specific suggestions.
Unknown to most, it is quite easy to directly show images in posts, hence the following general suggestion:
Please show images by using the Insert Photos button (looks like moon over mountains) for each at the top of the reply box which makes everything appear right there in your post together with your text, rather than he more conspicuous Drag&drop attachment which requires helpers to open a new tab for each image and wait for its showing, then go back and forth; and if they just click it and wait for its showing and press the X to get back to the text, the image is gone so they have to open it again and wait to see it again.
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AdChoices