How do you Find the exponential growth rate for a given data set?

Question

Christopher Agresta · Accepted Answer

The quick answer is: Take logs and find the slope.

If your data is exponential, then the log of the data will be linear. If the data set is exact, then the log of the data set will exactly fit a line with equation #y = mx + c# or similar. Slope #m# is given by the formula: #m = (Delta y)/(Delta x) = (y_2 - y_1)/(x_2 - x_1)# where #(x_1, y_1)# and #(x_2, y_2)# are two points on the line. For example, consider the data: #f(0) = 3# #f(2) = 12# #f(3) = 24# #f(7) = 384# Taking natural logs we find: #ln(f(0)) ~~ 1.0968# #ln(f(2)) ~~ 2.4849# #ln(f(3)) ~~ 3.17805# #ln(f(7)) ~~ 5.95064# Hence slope #~~ 0.693# and #f(x) ~~ 3 * e^(0.693 x)# #0.693# being the approximation of #ln(2)# that we found from the data. Actually #f(x) = 3*2^x = 3*e^(ln(2)x)# If the data set is inexact (e.g. measurements from an experiment), then take logs before performing a linear regression or similar.

Christopher Agresta · Answer

undefined To find the exponential growth rate for a given data set, you can follow these steps:

1. Determine if the data exhibits exponential growth. This is typically indicated by the data points increasing at a faster rate over time.

2. Plot the data on a graph, with time on the x-axis and the corresponding values on the y-axis.

3. Fit an exponential function to the data. This is usually done by using regression analysis or curve fitting techniques. The general form of an exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the base of the exponential function (growth rate), and $x$ is the time.

4. Once you have the equation for the exponential function, determine the value of $b$, which represents the exponential growth rate. This value indicates how much the quantity is growing or decaying over each unit of time.

5. Interpret the value of $b$ as the growth rate. If $b > 1$, it indicates exponential growth, while if $0 < b < 1$, it indicates exponential decay.