the assessed value can only go up a certain percentage per year unless some reason prompts a reassessment.
tax assessment on a property that was purchased many years prior will be less than market valuation.
this keeps people's property that has appreciated from being taxed at a rate they could no longer afford.
unfortunately, when an event (like a fire or hurricane) causes repairs to be needed that require a building permit, the property can be reassessed to the current market level.
this is why so many properties are sold after disasters. people will no longer be able to afford the property taxes.