Temporarily Color Roots
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I wish I could color every 3-4 weeks, but I'm afraid to color my hair more often than every 5-6 weeks. Is coloring hair more frequently than that damaging? How often can you safely color your hair?
What can you do about roots in the meantime? Can you rotate a permanent with a non-permanent dye? Are there any good products you can use to temporarily color roots? I apply a tinted toner called "Stay Red" to visible roots after washing my hair, but it's hard to apply. I wish they had a mascara-type applicator.
If the company that makes your permanent hair color also make a demi-permanent or semi-permanent color in the shade you use, you might try alternating between using the permanent and demi-permanent formulas.
Just be aware that since the demi-permanent color is designed to wash away in 6-12 shampoos, the longevity of the color will vary with your shampooing schedule. In addition, if your shampoo schedule is sufficient that the color lasts another 4-6 weeks, you'll see a larger new growth area at the scalp once your demi-permanent hair color begins to fade.
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There actually are products for retouching hair color, such as you describe, available at your beauty supply store. They are typically next to the hair color products. However, these tend only to be found in the basest colors and shades, so you may not want to risk unpleasant results from using them.
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See also:
Hair color application
How to touch up gray roots
How to make hair color last longer
The different types of hair color