Bleaching & Permanent Hair Color Change
A: No, bleaching won't change the color of your natural hair. Bleaching your hair only lightens the hair strand above the skin. Within a month of bleaching your hair, you'll notice a very prominent regrowth, where the hair has grown out again, being your natural color.
I think that you should keep in mind that bleaching hair can be an extreme change of hair color and appearance, especially if you have dark hair. The maintenance on bleached hair can be very high (especially when you have brown or dark brown/black hair), as you have to get your regrowth done every month.
If you don't maintain a bleached hairstyle very meticulously, you can end up in the awkward situation of walking around with a very unflattering solid regrowth, as the darker regrowth shows up in exaggerated difference in contrast to the bleached hair.
Bleaching your hair creates a "solid" bleached color, as all the hair is bleached at once. Getting highlights is a better idea as highlights are designed to blend in with your natural hair color, and the regrowth looks much better for it. Also, bleaching your whole head can be quite a risk if you don't have the proper knowledge and experience as to how to use the product.
Bleach also has the nasty habit of turning the hair brassy or yellow if you wash it off too soon, but if you leave it on for too long your hair will disintegrate. One should have the proper background knowledge of how to fix brassy or yellow hair with the correct toners and undertones.
I've known more than a handful of cosmetology students who broke their hair off two inches above the scalp because of over-bleaching their hair. And I see countless people daily who walk around with brassy/yellow hair, or hair colors and undertones that do not suit their complexions at all, and are not doing their appearance any favors.
In my opinion, bleach should be left to professionals or at least people who have a lot of experience from working with it. If you really want to experiment with color, I suggest trying hair color instead of a box of bleach. This would be safer for both your image and the health of your tresses.
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See also:
Hair bleaching problems
How does bleaching hair work?
Where does the color go when you bleach dark hair?