There are 7 types of logo:
Logo which consist of word only
e.g.: Sony, Panasonic, Toshiba
2. Geometric Typography
Logo which consist of word (brand) and combined with geometrical shape such as square, circle, elipse.
3. Initial Letter
Logo which consist of one or more letters as an initial.
Just like typographic logo (see no.1), but pictorial logo has stronger character since the typography was designed specially to create or design logo.
e.g.: Hard Rock Café, Coca Cola
Logo or symbol which has a direct associate with the product or main business core.
e.g.: Shell, 20th Century Fox
6. Allusive Logo
Logo that has indirect associate with the product or main business core.
e.g.: Mercedes Benz, Alitalia
Logo that has an abstract symbol. This kind of logo is usually hard to understand, because the design comes from a very long process. In fact, it’s losing the basic form. Since the logo becomes an allusive/abstract symbol, everybody would have different perceptions about this kind of logo.
Personally, I love an allusive and abstract logo, because logo must be unique, simple and have a strong character. The basic idea could be anything as long as related to the company. It could be the product, main business core, the philosophy, the background, geographic, vision or mission, etc.
If you’re designing logo for fitness and spa center, leave the muscle and fitness equipment kind of image away. Start to think deeper about the purpose of being healthy and fit instead. What people get when they do exercise and spa (regularly)? The answer is “Their bodies will be healthy and so do their souls”. That’s the main purpose of doing fitness and spa, get healthy body and soul equally. It means that you can illustrate this with something equal or balance. The most common symbol to illustrate the balance is a “Ying-Yang” symbol. But you can not use this symbol as it is (that wouldn’t be unique, would it?!). Try to modify and make it related to the fitness/spa field. That would be another challenge for you as a designer…
I’m sure that the exploration of each business area to be exposed on logo is unlimited.