There are several extensions of linear regression that apply to exponential growth and power law models. Problems 22-25 will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fi xed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fi xed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is y = αβx, where a and b are fixed constants to be estimated from data.
How do we know when we are dealing with exponential growth, and how can we estimate α and β? Please read on. Populations of living things such as bacteria, locusts, sh, panda bears, and so on, tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs (x, y) for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in y values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all y > 0.
Consider the following data, where x = time in hours and y = number of bacteria in a laboratory culture at the end of x hours.
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