Exponential functions are prevalent in many problems in science, business, economics, medicine, and sociology. One important function, the logistic function, is often used to analyze population growth that is limited by natural environmental factors. The general form of the logistic function is
where constants , , and are usually determined experimentally.
Suppose that population in a town is modeled by
where is number of population and is time in year. Let be the time when the population growth rate begin to decline and be the time when the population reach percent of its limit, then can be expressed as . Determine the value of .