No, that is not necessarily true. The amount of the exemptions that are granted by each taxing entity is subtracted from the market value of your residence and the taxes are calculated on that "lower value."
In addition, when you turn 65, you may receive a "tax ceiling" for your total school taxes; that is, the school taxes on your residence cannot increase as long as you own and live in that home.
If you significantly improve your home (other than ordinary repairs and maintenance), tax ceilings can go up. For example, if you add a swimming pool, a garage, a storage building, and a room to your home, or significantly remodel your home, your tax ceiling can rise. It will also change if you move to a new home. The tax ceiling is set at the amount you pay in the year that you qualify for the senior citizen exemption. The county, city or special district may adopt a tax ceiling.