Yes, it will. If the color is all one tone, no problem. Choose the new color you want. Click the fill-with-color bucket. Then click on the areas of the old color. They will be filled with the new color.
Now if the color of your logo is a gradient (for example, varying from light blue to medium green blue), Paint can't help you - unless you like changing the color pixel by pixel. If that's the case, you can try opening the logo in PowerPoint or Word and adjusting the colors of the whole logo at once. Or try Picasa.
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