Can you post a screeshot of the logo to show us what you mean?
Do you have access to any other software like Illustrator/Photoshop that you could create this in?
Have you tried using .PNG instead? You should get a better end result.
EDIT: What kind of screen are you viewing this on? HD/4k..? That could have an effect.
If PhotoShop is available I'd recommend the following method:
1. Export the page with the logo to PDF (use the PDF/X-4 preset if available; you did not tell your version of InDesign).
2. Open the PDF as Smart Object (that's different to just open the PDF with the open command) in PhotoShop.
3. Crop and scale to your desire.
4. "Save for Web..." as JPEG from PhotoShop (that menu item changed its name in CC 2015)
Personally, I think it would be better to design a logo not in InDesign but in Illustrator.
If you have some raster versions, like for web, in mind you should proof it in Illustrator with pixel preview, snap to pixel grid would be a method to draw along the pixel raster, but it would work only in one size but it might be important for the smallest expected size.
For print purposes you should use a PDF/X-4 or AI anyway.
And if it is a logo, a JPG is the worst file type you could imagine as
- JPG is a raster image
- JPG does not support transparency in any form and
- JPG has a lossy file compression which will cause spotty areas and a dirty white.
In print purposes you should use PDF/X-4 vector logo, but when it comes to other purposes like web or office, I recommend to use PNG instead. Or, where it is possible SVG.
the OP wants to place the logo on a website.
Ok. For that purpose it would be better to create an SVG file (vector type). All modern browsers should support this.
You cannot export to SVG with InDesign, but one is able to save that from Adobe Illustrator.
Export to PNG would be my choice, if I had to use a pixel format.
Transparency can be maintained (depending on the version of InDesign).
Using the above PDF => PhotoShop method, also PNG for Web could be exported.
That would be my recommendation in this case. Since we don't know about the file size that should be achieved, it could be JPEG as well.
Totally agree with Willi on this. If you can use Illustrator to create the original logo, even if it is for web. This will keep the original file as vector and will make your life a lot easier going forward for printing and updates, etc.
If you can use Illustrator then create it and export either a .PNG or .SVG or both, use the .SVG and have the .PNG as the fall-back. This is good practice.
.PNG is what you want as it carries transparency, I would steer well clear of .JPEG if you can as you will always lose clarity due the the lossy compression that's comes with .JPEG format.
All the best.