# Population Growth Formula

## Formula

### Summary

Expression Description
The population at time .
The initial population, sometimes modeled as so the formula returns the percentage of growth.
The exponential function, where
The relative growth rate constant. If , the population is exponentially growing, if the population stays the same, and if the population is decaying or decreasing.
The time elapsed.

## Usage

The population growth formula returns the population at a time given an initial population size and a growth rate constant .

### Growth Rate Constant

In practice, the growth rate constant can be calculated from data. For example, say we have two population measurements and taken at time and . Given the data, it’s tempting to say that the growth rate is since the population doubled in unit of time. However, in reality, the growth rate is equal to .

We know that simplifies to be from the subtraction property of logarithmns.

Of course, if we picked any two points on this graph we should calculate the same growth rate constant.