Well, the initial idea needs to come from a talented brain.
And keep in mind, this most brilliant quote about a logo:
a logo is great, if you can draw it with a stick in the sand.
Another good advice: never try to combine too much elements in a logo. A window cleaner might ask for a logo with "euhh... well, a sparkling window of course, with some water speckles, and let's add a ladder, a oh yes a flying sponge, don't forget the company name, in 3D..."
That's not a logo - that's a circus. Good luck !
But when it comes to the stage of refining and varying it, Illustrator is the best software to craft your logo in all necessary and extra variations. How it looks in purely black and white, what happens on non-white backgrounds, in combinations with or without type, versions with scaled type, etc. And Illustrator is used to eventually prepare it for further use in digital and printed reproduction.
Designers do use some tricks to quickly whip up dozens of logos, like combining or countering shapes, letting type or shapes cut away another shape or type, switch contrasts where shapes intersect, use strong optical lines, etc. So they might just open up Illustrator and go ahead immediately, performing their creative talent and skills right onto the computerized canvas.
Examples of logos with negative space
Also try to keep your logo highly visible in a sea of other stuff. Anything can look good on a white screen or clean sheet of paper. Judge it imposed on a picture of a crowded shopping street, or in a crammed webpage with lots of banners. Does it still stand strong ? Set requirements for the space surrounding it !
Many logo's with multiple colors and gradients are very difficult too maintain in all environments and conditions. It's fun to achieve in Illustrator, but costs tons of money to use in print and physical applications (interior and exterior design, fleet branding, etc.) With less colors and no gradients, it's technically much easier to produce and probably turns out more recognizable.
And finally, a logo can go horribly wrong,
even if you throw the whole universe into it:
'Breathtaking' Document Reveals Pepsi's Logo is Pinnacle of Entire Universe