Is there a reason you are rasterising the logo by exporting it a JPG? Illustrator is a vector drawing program, which allows you to create resolution independent artwork. That means you can print it at any size you want without loss of quality. Also what application are you printing the JPG from? You can: Print artwork straight from Adobe Illustrator: File > Print. Alternatively save the Illustrator logo as a PDF, and print it from Adobe Acrobat or Acrobat Reader. When exporting as a JPG and selecting/setting the resolution, you don't change the physical print-dimensions (wxh) of the artwork, instead you create a version that contains more pixels on an inch as you increase the resolution during the export. For example: a 10 x 10 cm document, exported at 300 ppi (pixels per inch), will end up being a 10 x 10cm document at 1182 x 1182 pixels. However more pixels doesn't change the physical print-dimension, it will still be 10 x 10 cm. You can check that using Adobe Photoshop for example. Open the JPG in Photoshop and choose Image > Image Size, and set the Width and Height units of measurement to cm or mm. OR if you were to place (File > Place, then click with the loaded graphics icon on the page) this JPG image into InDesign at 100% of the size, it'd still be 10 x 10 cm. I hope this helps clarify things.
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