The Scales Of e

"If you understand exponentials, the key to many of the secrets of the Universe is in your hand." - Carl Sagan's Billions And Billions (1998), p.23 

Introduction

This article extends the Scales Of 70 to demonstrate an exponential law of growth  that I call the Scales Of e.  This relates to the transcendental number e (approx 2.71818), which has been called the black jewel of the calculus (Berlinski, 1995), the magic number of growth (Tognetti, 1998) and the mystery number (Elwes, 2007).

It is easier to picture the Scales Of 70 as a weighing scale, with the rates for years of positive growth on one side, and the rates for years of negative growth on the other. The Scales of e uses a similar concept, except that the natural logarithms of the rates for years of positive growth are accumulated on one side, and the natural logarithms of rates for years of negative growth are accumulated on the other side.

Figure 1. The Scales Of e. For a given factor F, if the totalled Natural Logarithms of the positive growth rates exceed ( "outweigh") the totalled Natural Logarithms of the negative growth rates by Ln(F) then the population under consideration will multiply by F. If the totalled Natural Logarithms of the negative growth rates exceed ("outweigh") the totalled Natural Logarithms of the positive growth rates by Ln(F) then the population under consideration will divide by F.

Regardless of the growth rates used (or whether or not the growth rate varies), and regardless of the size of population, the Scales Of e always apply to all populations of all species at all times. This is why demographers and other scientists are able to use the Exponential Method so universally.

Exponentials and e

To denote any number as the exponent applied to the base e, this is expressed here as the EXP (number) function commonly used in applications such MS Excel. Here are some examples, using values from MS Excel:

EXP (1) =  e¹ = 2.718281828

EXP (2) = e² = 2.718281828 x 2.718281828 = 7.389056099

EXP (3) = e³ = 2.718281828 x 2.718281828 x 2.718281828 = 20.08553692

Natural Logarithms and e

Just as division and multiplication are inverse functions (the action of one undoes the other),  so the Natural Logarithm of a number is the inverse of EXP (number). This is expressed as LN (number). Here are some examples:

LN(1) = 0

LN(2) = 0.693147181 

LN(3) = 1.098612289

Multiplication and Division are Inverse Functions

Using Natural Logarithms to demonstrate that multiplication and division are inverse functions:

Table 1. LN(F) and LN(1/F) explored for the numbers 2, 3 and 4.

Adding Natural Logarithms is the same as multiplying real numbers. For example, the top row (Factor F = 2) is the equivalent of saying 2 * 1/2 = 1.

Note that LN(1/F) is always the negative of LN(F). Hence, LN(F) + LN(1/F) is always zero, regardless of what F is. However, this zero result is a Natural Logarithm and needs to be converted back to a real number by expressing it to the power of e.

e⁰ = 1, so this proves that multiplication and division are inverse functions as the net result of multiplying and dividing by any given factor is the equivalent of multiplying a number by 1.

LN and EXP are Inverse Functions

How do we know that LN and EXP are really inverse functions?

Table 2. LN and EXP explored for the numbers 1, 2 and 3.

In other words, any number can be expressed as LN(EXP(number)) or EXP(LN(number)). Thus Natural Logarithms really are Log (number)_e , the logarithm of a number in the base of e. 

The Scales Of 70 Revisited

As discussed in The Rule Of 70 and The Rule Of 72 Compared, and Rules Of Population, these rules of thumb are based on the Natural Logarithm of a factor. For doubling and halving, the factor is 2. Thus, LN(2) = 0.693147181.

A growth rate of 1%  can be expressed as 1.01. If it's negative growth, can be expressed as 0.99. The following two tables expresses growth rates in terms of their Natural Logarithms. 

Table 3 - Growth at positive, constant rates. For each period of growth for a given rate, the Natural Logarithm for that rate is added to a running total. Once this total exceeds LN(2), a starting population of any size will have doubled. LN(2) = 0.6931471806

Table 4 - Growth at negative, constant rates. For each period of growth for a given rate, the Natural Logarithm (which will a negative number) for that rate is added to a negative running total. Once this total exceeds LN(1/2)  - that is, exceeds LN(0.5) -  a starting population of any size will have halved. LN(0.5) = -0.6931471806

This is the reason why the Scales Of 70 works for constant positive and negative rates (as an approximation). However, the Scales Of 70 also works for variable positive and negative rates (as an approximation) whereas the Scales of e works for variable positive and negative rates (precisely). Before we explore a mixture of positive and negative variable growth rates for the Scales of e, let's examine an example of variable positive growth rates on their own:

Example 1: Variable positive population growth rates:

1 + 2 + 1 + 3 + 4 + 2 + 1 + 3 + 4 + 5 + 3 + 2 + 1 + 2 + 3 + 4 + 2 + 1 + 2 + 4 + 2 + 3 + 3 + 2 + 5 + 5  = 70

How many years did it take the population to double? Rather than adding the rates together, simply count how many rates there are - the answer is 26. In fact, it doesn't even matter which order these growth rates appear in, the result is always the same for the same set of growth rates.

Example 1 can be expressed as follows:

LN(1.01) + LN(1.02) + LN(1.01) + LN(1.03) + LN(1.04) + LN(1.02) + LN(1.01) + LN(1.03) + LN(1.04) + LN(1.05) + LN(1.03) + LN(1.02) + LN(1.01) + LN(1.02) + LN(1.03) + LN(1.04) + LN(1.02) + LN(1.01) + LN(1.02) + LN(1.04) + LN(1.02) + LN(1.03) + LN(1.03) + LN(1.02) + LN(1.05) + LN(1.05)  = LN(70)

Putting it all together

If there is a mixture of positive and negative rates, together with variable rates for both, then this is as complex as it can possibly get. The solution is revert back to the Scales Of e, with negatives on one side, and positives on the other. 

Example 2: This population will not quite double. Using the Scales Of e to measure the doubling time for mixed positive and negative variable growth rates. The sum of the growth rates can be calculated as 0.813627112 + -0.132106217 = 0.681520896 .  The elapsed time can be calculated as 23 + 6 = 29 (Black Periods + Red Periods = 29 periods). 

0.681520896 is less than 0.693147181 (which is LN(2)), so if this were a population then it would not quite double. How do we find what factor it does increase by?

EXP(0.681520896) = e^0.681520896 = 1.976882078 

Hence, the net result of these 29 periods of growth is that any starting quantity would be multiplied precisely by factor 1.976882078.

Handling Zero Population Growth

One last point is that a period of growth at 0% simply adds 1 to the period. 

Back to e

Recall that the EXP function is the inverse of the LN function. Taking values of LN from Table 3 and Table 4 above for the given growth rates, and expressing e to the power of those values, we have:

Table 5. The Scales Of 70 expressed in terms of e for positive constant rates, and negative constant rates. The whole number multiplier in the exponent indicates (according to the Rule Of 70 upon which the Scales Of 70 is based) the approximate doubling / halving period for a given rate. The final result I call the exponential factor, which precisely determines the multiplying factor to be applied (values of 0.5 are the equivalent dividing by 2).

For growth subjected to compound interest, these same exponential rules apply regardless of whether or not the rate of growth is constant (or fixed) or variable. 

The Scales Of e - Conclusion

Most people find it easier to think in terms of growth rates as percentages. Hence the 100 multiplier used below. 

For a given factor F, if the totalled Natural Logarithms of the positive growth rates exceed ( "outweigh") the totalled Natural Logarithms of the negative growth rates by 100 * Ln(F) then the population under consideration will multiply by F. If the totalled Natural Logarithms of the negative growth rates exceed ("outweigh") the totalled Natural Logarithms of the positive growth rates by 100 * Ln(F) then the population under consideration will divide by F. In both cases, take the absolute value (ignoring the sign). The multiplying or dividing period is always the total of the number of periods of growth.

The Scales of e explains why the Exponential Method works. Any combination of variable positive and negative compound interest growth rates can be applied to any sized starting quantity, and it is always possible to derive the exponential factor using the exponential rules mentioned above. The Scales of e is also something that is not meant to exist - a simple growth model that applies to all populations of all species at all times. The Scales of e can also be used to explain Exponential Brownian Motion, an inaptly named growth model for share price fluctuations.

Hence there is nothing, other than mathematical tradition, that requires exponential growth to be limited to a constant rate of growth (see my article What Is Exponential? for more on that subject). This clearly demonstrates the exponential nature of both fixed rate compound interest and variable rate compound interest. 

References

e: the mystery number - Richard Elwes, 18th July 2007 - from NewScientist.com (accessed 27/07/2007)

Back to Top

Send email to exponentialist@optusnet.com.au with questions or comments about this web site.
Copyright © 2001 David A. Coutts

Last modified: 05 June, 2009