Are you the person responsible for managing the direct relationship with the client? If so, your employer has not given you the necessary tools. As mentioned below, there should be timeline/budget considerations. Managing time/budget is critical to a projects success, for both sides of the relationship.
If the person managing this client/project is not you, they are the person to discuss this issue with, and see if you are still within the scope of the project. If you are being asked to manage this relationship with little or no guidance, there are a couple of issues. First, you shouldn't be considered an intern, and should receive proper wages. And second, you have not been given the scope information necessary to do your job.
I know this seems like a painful process, but get used to it. Learning how to manage these situations effectively is a crucial element to business relations.
I am in a internship too atm. And I can tell there have been changes here and then here too, and sometime it have been frustrating, but sometime needed I guess.
But 100 logos is a bit overkill I guess. I think you should talk about it indeed. Do not be afraid to discuss this, and also give your own thought what is good and bad, and why it is. Sometime people don't really know what they want, so sometime you have to lead them toward a design by sharing your thought about what is good and what is not.
A really good client knows what they want, explains it clearly, provides useful feedback, and is pleased when the project goals have been met.
A bad client may not know what they want, may not be able to explain it clearly, may simply want to "be in control" of the project but lack the knowledge or experience to be effective in that role, may keep changing their mind, may wander aimlessly hoping for that design that they'll "know when they see it", may make choices based on their good or bad mood instead of rational criteria. I've worked for a crazy client so I understand your pain.
It's always possible that they just didn't dig your style. That doesn't mean your choices were bad or their taste was bad. Sometimes project partners just aren't a good fit.
Although it's frustrating, it doesn't sound like it is affecting your pay. You could ask for your supervisors opinion. If the company feels that you could be better utilized on another project, they will make that decision.
Take it to your boss or senior designer first. Tell them the situation, ask them for advice or what they would do. If it sounds ridiculous to them, they'll know how to handle it.
It sounds to me like the client you're dealing with is one of these that thinks they want something fresh and new but deep down psychologically they just want the same old thing. The result is that they want something "new" but refuse anything that actually IS new.
I've seen people with that issue in all areas of business, it can be ridiculous.