Logos are a special case that reside in the creative domain covered by copyright, and the domain of trademarks related to identity. The logo itself, the style, colour, etc are usually a result of a creative process. This is usually what you can copyright. The logo denominating your product, company, etc, is what you can trademark.
So to prevent someone from using your logo to do anything besides use it as an identification tool, you would need a copyright for it. To prevent them to use it as an identifier you would need to register a trademark.
While it may seem that having a copyright would suffice, in my experience a copyright would not cover the case where they would use the logo to promote a product based on your original product (which held the logo). Some copyright licenses permit this but I would suggest you carefully pick one that does allow exactly what you want to allow.
Note that I assumed you can actually copyright your logo which may not be possible as the creative process might not be unique enough. If for some reason a copyright is not applicable to your logo, you should at least register the trademark to prevent reuse.
To sum up and answer your final point; even with a copyright, should it apply to your logo, others can use it for other purposes, likely even purposes that are completely unrelated to your activity.