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---September 5, 1990---
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One of the important characteristics of environmental problems is the way they're growing. It is important for the public (and for news reporters and environmentalists) to understand growth. This is often easy to do because of the way most things grow. As Ralph Lapp has made clear in his book, THE LOGARITHMIC CENTURY, human population (and most of the things related to humans such as automobiles, chemicals and chemical wastes), are growing exponentially. This permits us to make accurate growth projections easily.

First a definition: A quantity is growing exponentially if it grows by a fixed percentage of the whole in a fixed time period. A familiar example of a quantity that is growing exponentially is a bank account that grows at 6% per year; it grows by a fixed percentage of the whole (6%) in a fixed period of time (a year).

There are some rules about exponential growth that allow us to make quick and accurate projections into the future.

RULE 1: To determine the doubling-time (d) for an exponentially-growing quantity, divide the annual percentage rate of increase (p) into 70.

d = 70/p [Rule 1]


d = the time it takes for the quantity to double in size;

p = the annual increase expressed as a percentage.

Thus the savings account growing at 6% per year is doubling every 70/6 = 11.7 years. Thus $5 growing at 6% per year will grow to $10 in 11.7 years. By the same reasoning, a quantity that is growing at 10% per year--such as production of a chemical--will have a doubled annual production rate in 7 years. (For those who are curious, 70 is used because it is very close to 100 times the natural logarithm of 2, which is 0.693.)

RULE 2: If we know the doubling time for an exponentially growing quantity we can calculate the annual percentage increase (p) by using a variation of Rule 1.

p = 70/d [Rule 2]


d = the time it takes for the quantity to double in size;

p = the annual increase expressed as a percentage.

If we are told that something is doubling in 5 years, we know that it is growing at 70/5 = 14% per year.

RULE 3: The fundamental equation for exponentially growing quantities is:

N_sub_t = N_sub_o*e**kt [Rule 3]


N_sub_o is some original amount;

N_sub_t is the amount that it has grown to at some later time, t;

e is a constant, equal to 2.718 (it is the base of natural logarithms);

k = the annual percentage increase expressed as a decimal fraction (in other words, it's the value we've been calling p, divided by 100);

t = time (in any units you care to choose).

Don't be put off by the strange notation; N_sub_o is pronounced "N sub O" and N_sub_t is pronounced "N sub T." This is the way mathematicians and physicists like to talk about quantities, but once you get used to the odd way of expressing them, the ideas themselves are simple enough. To handle the arithmetic involved in such an equation, remember that when two items are written next to each other, it means that they should be multiplied together. In this example, k and t have been written kt and this means that k is multiplied by t. (We have also used an asterisk to indicate that two numbers should be multiplied by each other, so kt and k*t mean the same thing--multiply k times t.)

There is a standard order in which mathematical operations are carried out. First, any exponents should be evaluated (figured out). In this case, kt is an exponent, so you multiply k times t first. Next you carry out the exponentiation: in this case, you raise e to the power of k*t. (A $15 scientific calculator from Radio Shack can raise e to any power for you.) Next you carry out any multiplication or division; in this case, because they are written next to each other, you would multiply N_sub_o times whatever you got when you raised e to the power of kt. Last, you do any addition or subtraction; in this particular example there isn't any addition or subtraction indicated.

Parentheses are used to change the order in which mathematical operations are carried out; always do what's inside parentheses first. Start inside the innermost parentheses and work your way outward.

Example of Rule 3: If production of hazardous wastes is growing at 6.5% per year [thus doubling every 10.8 years] and if we produced 30 million tons of hazardous waste in 1980, how much hazardous waste will we be producing in 1995? N_sub_o = 30 million tons; t = 1995-1980, or 15; k = 6.5/100, 0.065. Therefore, N_sub_t (the amount of waste produced at time t), when t = 15, is e raised to the power of (0.065 x 15, or 0.975), times 30 million. Using a scientific calculator, we raise e to the power of 0.975 and we get 2.65. Therefore, the amount of waste to be produced in 1995 = 30 million tons times 2.65, or 79.5 million tons, assuming that the growth-rate continues to average 6.5% per year between 1980 and 1995.

RULE 4: If a quantity is growing exponentially, during one human lifetime (assumed to be 70 years) it will grow by a factor of 2 raised to the power of p, where p is the annual percentage rate of increase. (The phrase "it will grow by a factor of" means "its growth can be calculated by multiplying by.")

N_sub_t after 70 years = N_sub_o*2**p [Rule 4]


N_sub_o is some original amount;

N_sub_t is the amount that it has grown to at some later time, t;

p = the annual increase expressed as a percentage.

Table 1 gives 2p for many typical values of p.

Thus when we say that production of chemical X is increasing at 10% per year, we can calculate that during one human lifetime the annual production rate of chemical X will increase by a factor of 2**10, or 1024. That is to say, if we produced 1,000,000 (one million) pounds of chemical X in 1980 and our production is growing at 10% per year, at the end of one human lifetime we will be producing 1,000,000 x 1024 = 1,024,000,000 (or more than one billion) pounds of chemical X annually.

At this point we should make the distinction between predictions and projections. A prediction is a statement of what someone thinks is going to happen. A projection is a statement of what will happen if things don't change. As we are using the term here, a projection is based only on the past record of the growth of something. A prediction may take into consideration many other factors besides the past record of the growth of something; for example, a prediction may take into account how we humans are likely to react to a scary projection of future growth. A projection can--by itself--make things change. (In other words, a projection may cause us to change our predictions.) Thus one is not predicting that we will increase our production of some chemical by a huge amount during one lifetime. One is simply projecting that--based on past growth records--such future growth will occur unless something changes. Sometimes the frightening implications of growth projections are--by themselves--sufficient for people to see that we've got to slow down some rate of growth.

(of anything)
150                                    *
.     Typical Curve Produced By
.     Exponential Growth               *
125   (Growth rate = 10% per year)
.                                     *
100                                 *
.                                 *
.                              *
50                        *
.                   *
.             *
.* * *  *
0 _______________________________________
1940   1950   1960   1970   1980   1990
.              TIME

Figure 1. Typical curve created by something growing exponentially. Notice that at the beginning, the curve is not rising steeply; as time passes, however, the curve becomes steeper and steeper. The larger the quantity becomes, the faster it grows; this is the main characteristic of things that grow exponentially.


Table 1. Various Powers of 2
If p equalsThen 2^p equals
Table 1. The value of 2 raised to the power of various annual rates of increase, p, expressed as a percentage.

--Peter Montague, Ph.D.

Descriptor terms: mathematics; predictions; exponential growth;

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