Up Rules Of Population Rule Of 70 and Rule Of 72 Compared The Scales Of e External Links: Malthusian Growth Model - by Steve McKelvey. Mathematical Modelling in a Real and Complex World - by the Connected Curriculum Project Exponential Growth and The Rule Of 70 - by EcoFuture  e the EXPONENTIAL - the Magic Number of GROWTH - Keith Tognetti, University of Wollongong, NSW, Australia The Number e - A history of the number e from St Andrews University, Scotland Ivar's Peterson's Mathtrek - Hunting e An Intuitive Guide To Exponential Functions & E - Better Explained Courage with 2.718281828 ~ e - Boris Reitman The Scales Of e

"If you understand exponentials, the key to many of the secrets of the Universe is in your hand." (Carl Sagan 1997)

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:

 Factor F LN(F) LN(1/F) LN(F) + LN(1/F) 2 0.693147181 - 0.693147181 0 3 1.098612289 -1.098612289 0 4 1.3862943611 -1.3862943611 0

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?

 1 2 3 Exp(1) = 2.718281828 Exp(2) = 7.389056099 Exp(3) = 20.08553692 LN(1) = 0 LN(2) = 0.693147181 LN(3) = 1.098612289 LN(EXP(1)) = 1 LN(EXP(2)) = 2 LN(EXP(3)) = 3 EXP(LN(1)) = 1 EXP(LN(2)) = 2 EXP(LN(3)) = 3 e0 = 1 e0.693147181 = 2 e1.098612289 = 3

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.

 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Period LN(1.01) LN(1.02) LN(1.03) LN(1.04) LN(1.05) LN(1.06) LN(1.07) LN(1.08) LN(1.09) LN(1.1) 1 0.009950331 0.019803 0.029559 0.039221 0.04879 0.058269 0.067659 0.076961 0.086178 0.09531 2 0.019900662 0.039605 0.059118 0.078441 0.09758 0.116538 0.135317 0.153922 0.172355 0.19062 3 0.029850993 0.059408 0.088676 0.117662 0.14637 0.174807 0.202976 0.230883 0.258533 0.285931 4 0.039801323 0.079211 0.118235 0.156883 0.195161 0.233076 0.270635 0.307844 0.344711 0.381241 5 0.049751654 0.099013 0.147794 0.196104 0.243951 0.291345 0.338293 0.384805 0.430888 0.476551 6 0.059701985 0.118816 0.177353 0.235324 0.292741 0.349613 0.405952 0.461766 0.517066 0.571861 7 0.069652316 0.138618 0.206912 0.274545 0.341531 0.407882 0.473611 0.538727 0.603244 0.667171 8 0.079602647 0.158421 0.23647 0.313766 0.390321 0.466151 0.541269 0.615688 0.689422 0.762481 9 0.089552978 0.178224 0.266029 0.352986 0.439111 0.52442 0.608928 0.692649 0.775599 10 0.099503309 0.198026 0.295588 0.392207 0.487902 0.582689 0.676586 0.76961 11 0.109453639 0.217829 0.325147 0.431428 0.536692 0.640958 0.744245 12 0.11940397 0.237632 0.354706 0.470649 0.585482 0.699227 13 0.129354301 0.257434 0.384264 0.509869 0.634272 0.757496 14 0.139304632 0.277237 0.413823 0.54909 0.683062 15 0.149254963 0.297039 0.443382 0.588311 0.731852 16 0.159205294 0.316842 0.472941 0.627531 17 0.169155625 0.336645 0.5025 0.666752 18 0.179105955 0.356447 0.532058 0.705973 19 0.189056286 0.37625 0.561617 0.745194 20 0.199006617 0.396053 0.591176 21 0.208956948 0.415855 0.620735 22 0.218907279 0.435658 0.650294 23 0.22885761 0.45546 0.679852 24 0.23880794 0.475263 0.709411 25 0.248758271 0.495066 0.73897 26 0.258708602 0.514868 27 0.268658933 0.534671 28 0.278609264 0.554474 29 0.288559595 0.574276 30 0.298509926 0.594079 31 0.308460256 0.613881 32 0.318410587 0.633684 33 0.328360918 0.653487 34 0.338311249 0.673289 35 0.34826158 0.693092 36 0.358211911 0.712895 37 0.368162242 38 0.378112572 39 0.388062903 40 0.398013234 41 0.407963565 42 0.417913896 43 0.427864227 44 0.437814558 45 0.447764888 46 0.457715219 47 0.46766555 48 0.477615881 49 0.487566212 50 0.497516543 51 0.507466874 52 0.517417204 53 0.527367535 54 0.537317866 55 0.547268197 56 0.557218528 57 0.567168859 58 0.577119189 59 0.58706952 60 0.597019851 61 0.606970182 62 0.616920513 63 0.626870844 64 0.636821175 65 0.646771505 66 0.656721836 67 0.666672167 68 0.676622498 69 0.686572829 70 0.69652316

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

 -1% -2% -3% -4% -5% -6% -7% -8% -9% -10% Period LN(0.99) LN(0.98) LN(0.97) LN(0.96) LN(0.95) LN(0.94) LN(0.93) LN(0.92) LN(0.91) LN(0.9) 1 -0.010050336 -0.0202 -0.03046 -0.04082 -0.05129 -0.06188 -0.07257 -0.08338 -0.09431 -0.10536 2 -0.020100672 -0.04041 -0.06092 -0.08164 -0.10259 -0.12375 -0.14514 -0.16676 -0.18862 -0.21072 3 -0.030151008 -0.06061 -0.09138 -0.04242 -0.15388 -0.18563 -0.21771 -0.25014 -0.28293 -0.31608 4 -0.040201343 -0.08081 -0.12184 -0.0032 -0.20517 -0.2475 -0.29028 -0.33353 -0.37724 -0.42144 5 -0.050251679 -0.10101 -0.1523 0.036018 -0.25647 -0.30938 -0.36285 -0.41691 -0.47155 -0.5268 6 -0.060302015 -0.12122 -0.18276 0.075239 -0.30776 -0.37125 -0.43542 -0.50029 -0.56586 -0.63216 7 -0.070352351 -0.14142 -0.21321 0.11446 -0.35905 -0.43313 -0.50799 -0.58367 -0.66017 -0.73752 8 -0.080402687 -0.16162 -0.24367 0.15368 -0.41035 -0.495 -0.58057 -0.66705 -0.75449 -0.84288 9 -0.090453023 -0.18182 -0.27413 0.192901 -0.46164 -0.55688 -0.65314 -0.75043 -0.8488 10 -0.100503359 -0.20203 -0.30459 0.232122 -0.51293 -0.61875 -0.72571 -0.83382 11 -0.110553694 -0.22223 -0.33505 0.271342 -0.56423 -0.68063 -0.79828 12 -0.12060403 -0.24243 -0.36551 0.310563 -0.61552 -0.7425 13 -0.130654366 -0.26264 -0.39597 0.349784 -0.66681 -0.80438 14 -0.140704702 -0.28284 -0.42643 0.389005 -0.71811 15 -0.150755038 -0.30304 -0.45689 0.428225 -0.7694 16 -0.160805374 -0.32324 -0.48735 0.467446 17 -0.17085571 -0.34345 -0.51781 0.506667 18 -0.180906045 -0.36365 -0.54827 0.545887 19 -0.190956381 -0.38385 -0.57872 0.585108 20 -0.201006717 -0.40405 -0.60918 21 -0.211057053 -0.42426 -0.63964 22 -0.221107389 -0.44446 -0.6701 23 -0.231157725 -0.46466 -0.70056 24 -0.24120806 -0.48486 -0.73102 25 -0.251258396 -0.50507 -0.76148 26 -0.261308732 -0.52527 27 -0.271359068 -0.54547 28 -0.281409404 -0.56568 29 -0.29145974 -0.58588 30 -0.301510076 -0.60608 31 -0.311560411 -0.62628 32 -0.321610747 -0.64649 33 -0.331661083 -0.66669 34 -0.341711419 -0.68689 35 -0.351761755 -0.70709 36 -0.361812091 -0.7273 37 -0.371862427 38 -0.381912762 39 -0.391963098 40 -0.402013434 41 -0.41206377 42 -0.422114106 43 -0.432164442 44 -0.442214778 45 -0.452265113 46 -0.462315449 47 -0.472365785 48 -0.482416121 49 -0.492466457 50 -0.502516793 51 -0.512567129 52 -0.522617464 53 -0.5326678 54 -0.542718136 55 -0.552768472 56 -0.562818808 57 -0.572869144 58 -0.58291948 59 -0.592969815 60 -0.603020151 61 -0.613070487 62 -0.623120823 63 -0.633171159 64 -0.643221495 65 -0.65327183 66 -0.663322166 67 -0.673372502 68 -0.683422838 69 -0.693473174 70 -0.70352351

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.

 Rates for periods of negative growth Rates for periods of positive growth LN(0.99) + LN(0.97) + LN(0.98) + LN(0.95) + LN(0.99) + LN(0.99) LN(1.01) + LN(1.02) + LN(1.02) + LN(1.02) + LN(1.02) + LN(1.03) + LN(1.04) + LN(1.04) + LN(1.0 4) + LN(1.04) + LN(1.05) + LN(1.05) + LN(1.06) + LN(1.05) + LN(1.05) + LN(1.0 4) + LN(1.03) + LN( 1.03) + LN(1.03) + LN(1.03) + LN(1.05) + LN(1.05) + LN(1.0 3) Total -0.132106217 for 6 periods Total 0.813627112 for 23 periods

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) = e0.681520896 = 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:

 1% 2% 5% 7% e(0.009950331 * 70)  = e0.69652316 =2.006763369 e(0.019803 * 35)  = e0.693092 =1.999889642 e(0.04879 * 14) = e0.683062 =1.979931009 e(0.067659* 10) = e0.676586 =1.967150404 e(-0.010050336 * 70) = e-0.70352351 =0.494838659 e(-0.0202 * 35) = e-0.70709 =0.493076966 e(-0.05129 * 14) = e-0.71811 =0.487673088 e(-0.07257* 10) = e-0.72571 =0.483980821

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

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

Sagan, Carl. Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium. Headline Publishing, 1997