Convert a jpg logo ro a vector

Make your graphic using pen and shape tools then go to  File > Export > Export As  and choose SVG as the export format.

Dave

I would export to Illustrator where you can convert bitmap to vector more easily.

What matter most is what is inside your image :

• How many colors? ( for print it will be CMYK, so if you used RVB the conversion will often change the colors)
• Typography used (You can convert the text yo vector)
• Gradients
• Shapes Complexity

…

as well Photoshop is not the tool for print brand identity. But maybe you can get out of it anyhow.

Show us yiour logo so wze can evaluate the issue.

Photoshop is not a tool for saving vectors. It's for photos. Wrong tool, even though it can do some vector additions to a photo, it's not a vector result.

Use vector shapes or type layers as already suggested. If you want easily to convert existing pixel based logo to vector then you must redraw it using Pen tool or import in Illustrator and try there using Trace feature which can help to automate process.

With a Custom shape you can scalle its size well but yoy only get fill.  A layer style cans be added  you can de wonders with a layer  style these days.