Rainbow is a universally positive symbol that is associated with a positive outlook on life and abundance. If your brand promises to bring people happiness and offers variety you can consider such a colorful logo.
At the time when Apple originally introduced the rainbow colored logo it was criticized because it was hard to reproduce on various surfaces, but today printing technology evolved significantly so a multi-colored logo should not pose any problems especially on screen.
It's important to remember that such range of bright colors attracts attention so it's great for outdoor use if you want to draw in customers, but could be disturbing is used in print inappropriately where you want to keep your reader's attention on content. It's also advisable to ask a designer to prepare a single color version of your logo for those occasions where color reproduction is not possible.
Here is what Wikipedia writes about the mythology of the rainbow as a symbol: "The rainbow, a natural phenomenon noted for its beauty and inexplicability, has been a favorite component of mythology throughout history. The Norse saw it as Bifrost; Judeo-Christian traditions signs it as a covenant with God not to destroy the world by means of floodwater. Finding a mythology that does not include the rainbow somewhere may be the true challenge. Whatever the culture or continent, our species' earliest rainbow is the rainbow of the imagination. Whether as bridge, messenger, archer’s bow, or serpent, the rainbow has been pressed into symbolic service for millennia. The myriad rainbow bridges and myths built by the world’s peoples clearly tell us more about human hopes and fears than they do about nature’s rainbow."
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