IntroductionThe material below explains the concepts of exponential growth and doubling time  the time it would take a population to double if it were growing exponentially at a constant rate. Throughout human history, human population has generally grown in an exponential manner, and discussion of exponential doubling times has been appropriate (see chart at the end of this page).Now, the dynamics of population growth are more complex and growth can no longer be considered truly exponential. The issues of the 21st Century are migration, aging, the youth bulge, urbanization, and all of the very new and very serious socioeconomicpolitical consequences, as well as continued growth in the developing world and, of course, pressures on resources and the environment  global and local. This does not mean that countries will not double their population. Many will. For example, the U.S. is projected to double its population this century, practically within the lifetimes of today's children. So it is still appropriate to reference the time in which these countries will double their population, which is perhaps the most illuminating manner of presenting population growth to the lay person. But it should be noted that the explanations below refer to a constant rate of growth, which is not as applicable to countries now as in the past. Exponential GrowthA quantity grows exponentially when its increase is proportional to what is already there. A common example is compound interest, where $100 invested at 7% per year annual compound interest will double in 10 years! Exponential growth applies to populations, too  if a population grows at 7% per year, it, too, will double in 10 years.There are surprising consequences to the phenomenon of exponential growth. The $100 invested at a 7% annual return will double in 10 years to approximately $200, double in another 10 years to approximately $400, and double again in the next 10 years to approximately $800. Significant gains can be made by simply relying on exponential growth over time. One way of saying this is that the longer you wait (on your investment), the faster your returns come in. In the following graph, you can see that over time, returns increase dramatically. 

Unfortunately, exponential growth works against us, too. When populations continue to grow, the impact of growth becomes increasingly significant over time. In other words, because of the nature of exponential growth, "when things get bad, they get bad in a hurry". Consider a country with 100 people, growing at 7% per year. In 10 years, the population will double to 200 people, in another 10 years it will double again to 400 people, and ten years after that it will double again to 800 people. The following graph shows this exponential population growth. Notice the shape of the curve is the same as in the graph above! 

The Lily Pond Parable
The Rule of 70The Rule of 70 is useful for financial as well as demographic analysis. It states that to find the doubling time of a quantity growing at a given annual percentage rate, divide the percentage number into 70 to obtain the approximate number of years required to double. For example, at a 10% annual growth rate, doubling time is 70 / 10 = 7 years.Similarly, to get the annual growth rate, divide 70 by the doubling time. For example, 70 / 14 years doubling time = 5, or a 5% annual growth rate. The following table shows some common doubling times:
Math Behind the Rule of 70The use of natural logs arises from integrating the basic differential equation for exponential growth: dN/dt = rN, over the period from t=0 to t = the time period in question, where N is the quantity growing and r is the growth rate.
where N(t) is the size of a quantity after t intervals have elapsed, N(0) is the initial value of the quantity, e is the base of the natural logarithm, r is the average growth rate over the interval in question, and t is the number of intervals. Natural logarithms (that is, logarithms to the base e) come in from this integration. Natural logs are sometimes abbreviated ln to distinguish them from "common" logarithms of base 10. In our case, t is usually given in years and r is the average annual growth rate. However, the formula works for seconds, weeks, centuries, etc. The ratio of the final to the initial values of N is N(t)/N(0), and is equal to e^{rt} power. (Dividing through by N(0).) The natural logarithm of the ratio N(t)/N(0) is equal to rt. (Taking the natural log of e raised to a power is simply the power itself.) The growth rate r is then given by natural log N(t)/N(0) divided by t.
If one knows the growth rate as a decimal fraction, then the doubling time t_{2} = 0.69 / r. If the growth rate is given in percent, then 0.69 must be multiplied by 100, and the doubling time = 69/r. This is the origin of the rule of 70, i.e., 69 is rounded up to 70. For the results to be accurate, all of these calculations assume that the growth rate remains unchanged throughout the interval in question, that is, that the growth is exponential at the average rate for the entire period. Most quantities don't really grow that way (compound interest is an exception), so this method is generally an approximation for the real world. When considering growth over a period of years, it is important to note that taking the natural logarithm of the ratio of the final value to the initial value and dividing by the time period in years gives the average annual growth rate. For example, metro Denver population grew 15% from 1990 to 1996. The simple arithmetic average of growth is 2.5% per year (15% / 6 years = 2.5%/year). Strictly speaking the rule of 70 applies to exponential growth, which means that the compound average population growth rate must be divided into 70 to get the doubling time. The compound average growth rate involves natural logarithms. To find the compound growth rate, take the natural log of the ratio of the final to the initial populations and divide this by 6 years. 15% growth means that the ratio of final to initial populations is 1.15. (The final population is 115% of the initial population, considered to be 100%; 115% / 100% = 1.15.) The natural log of 1.15 is 0.14. Dividing 0.14 by 6 years = 0.023/year, or an average geometric increase of 2.3% / year. Finally, divide 70 by 2.3 to give a doubling time of 30 years.  Mathematical explanation by Dick Schneider Additional ReferencesAlgebra the Easy Way, Douglass Downing; 1996, ISBN 0812093933, (334p, $11). Includes a basic discussion of exponential functions and logarithms. Ask. Dr. Math Beyond the Limits, Confronting Global Collapse, Envisioning a Sustainable Future, Donella Meadows. The first two chapters present an excellent discussion of exponential growth. Doubling Times and the Rule of 70, Mathematical Methods for Economics, Michael Klein. Is There A Population Problem?, Albert A. Bartlett. Population Doubling Time, PopulationEnvironment Balance. Answers to The Lily Pond Parable
"Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist" Kenneth Boulding 

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Copyright 19952003 Fred Elbel. This material may be freely used and distributed only for noncommercial purposes, with credit. Trademarks and copyrighted items remain the property of the owner. 