It would be quite incorrect to assume that baking soda and deodorant are substitutes for each other or even complimentary to each other. Applying baking soda to the skin is however useful as an abrasive in order to exfoliate the skin. Apart from this, applying baking soda to the skin is not advisable, as it would disturb the pH balance of the skin.
The reason why your armpits appear black is probably hair removal over time that has caused the hair to become a bit coarser. When you shave the hair of your armpits, the residual hair under the surface of your skin creates the illusion – especially from a distance, that the skin has turned black. If you look closely at the skin in the area, you will notice that in fact the skin is not dark at all. You can try to modify your shaving technique to the way men shave their facial hair. First, prepare the area for shaving by soaping the area with warm water and “massaging the water” into the skin. Next, take a double or triple blade razor and shave in one direction, then shave back against any grain remaining. Keep this up until you feel absolutely no friction. You will notice that the black shade would have decreased. If this change in your technique of shaving does not work, you can try an epilating technique like using a hair removal cream. Hair removal creams might feel a bit unpleasant to use and might cause a burn in the area if you have sensitive skin. When the hair follicle is gone, then there is nothing under the skin left to cause the shading as would normally be seen. Unfortunately using these creams on a long term basis too can aggravate the skin and cause darkening. Waxing is a painful treatment but since it removes the hair from its roots, it does not cause the skin to appear darker.
There are times that you would probably have to exfoliate you skin in the region since you clearly do not allow any hair to grow in the area. Usually, having some hair in the armpit causes an exfoliation because of the friction between the hair and the skin. The exfoliation mechanism you should use then is salicylic acid. This is an acid that is quite easy on the skin and has the ability to break down the protein bond that holds dead skin together, causing it to fall off easily. Apply this, instead of the baking soda, try sugar, and rub it in circles around the area.