There are several extensions of linear regression that apply to exponential growth and power law models. We will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fixed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fixed 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 α and β 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, fish, 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. x 1 2 3 4 5 y 3 12 22 51 145 (b) Now consider a transformation y' = log y. We are using common logarithms of base 10 (however, natural logarithms of base e would work just as well). x 1 2 3 4 5 y' = log y 0.477 1.079 1.342 1.708 2.161 (c) Use a calculator with regression keys to find the linear regression equation for the (x, y) data pairs with sample correlation coefficient r = 0.882. (Round your answers to three decimal places.) ŷ = _+_ x (d) Use a calculator with regression keys to find the linear regression equation for the (x, y') data pairs with sample correlation coefficient r = 0.993. The sample correlation coefficient r = 0.882 for the (x, y) pairs is not bad. But the sample correlation coefficient r = 0.993 for the (x, y') pairs is a lot better! (Round your answers to three decimal places.) y' = _+ _x (e) The exponential growth model is y = αβx. Let us use the results of part (d) to estimate α and β for this strain of laboratory bacteria. The equation y' = a + bx is the same as log y = a + bx. If we raise both sides of this equation to the power 10 and use some college algebra, we get y = 10a(10b)x. Thus, α ≈ 10a and β ≈ 10b.Use these results to approximate α and β and write the exponential growth equation for our strain of bacteria. (Round your answers to three decimal places.) α = β = y = · x