The carrying capacity formula is a mathematical expression for the theoretical population size that will stabilize in an environment and can be considered the maximum sustainable population. The formula is essentially a mathematical way to provide a limit to the otherwise exponential growth of a species.
The difference between an exponential and logistic growth model is evident when looking at a graph of the two populations over time. The exponential model, one that does not consider carrying capacity, will grow exponentially forever. However, the logistic model, which is a model of populations that does consider the carrying capacity, grows rapidly and then approaches the carrying capacity asymptotically. As a result, after the population reaches its carrying capacity, it will stop growing and the number of births will equal the number of deaths.