Certain colors, especially vibrant ones (violetes, oranges and greens), will not look good when converted to CMYK. The reason for that is the fact that when combining red, green and blue in a digital space, you can get a much wider color combination than with analogue printing process of combining cyan, magenta, yellow and black.
How larger RGB color space is than CMYK you can easily see on a diagram bellow (CMYK is shown inside the purple lines). Everything outside that space will be showh with the colors from within the CMYK color space (you can slightly improve the results by changing the rendering intent):
So, to get the best results, make sure you`re converting to a correct color space, because that can increase the gamut if chosen correctly (Fogra 39 for Europe and SWOP v2 for USA are most common ones).
For best results when printing a logo, choose a Pantone color from PMS color chart.
If you do not have a possibility of working with Pantone, you can make conversion situation a little better by removing colors from unnecessary channels - for example, if you have a bright green, thy removing magenta or adding it`s information to another channel in channel mixer, it will look brighter.