

Rules Of Population
Introduction
Much more obscure than the Rule Of 70 are the Rule Of 110 (for trebling and thirding) and the Rule Of 140 (for quadrupling and quartering). Other rules exist for other factors, but these two will be sufficient to confirm some natural patterns in nature.
Rule Of 110
Taking a growth rate of r, the accurate value X is calculated as:
X = Ln 3 / Ln (1 + (r / 100))
Taking a growth rate of r, the accurate value Y is calculated as:
Y = Ln 3 / Ln (1  (r / 100))
Curiously enough this difference seems to equate to the Natural Logarithm of 3 (Ln 3 = 1.098612), which is where we get the Rule Of 110 in the first place. This confirms a similar finding for the Rule Of 70.
Ln 3 = X + Y
This can be written:
X = ABS( Y) + Ln 3
Trebling Period  Thirding Period  X + Y  
Growth Rate  Rule Of 110 
Accurate Value (X) 
Rule Of 110 
Accurate Value (Y) 

1  110  110.409624  110  109.3110026  1.098621 
2  55  55.47810764  55  54.37945872  1.098649 
3  36.66666667  37.16700967  36.66666667  36.06831495  1.098695 
4  27.5  28.01102276  27.5  26.91226388  1.098759 
5  22  22.51708531  22  21.41824388  1.098841 
6  18.33333333  18.85417668  18.33333333  17.75523427  1.098942 
7  15.71428571  16.23757367  15.71428571  15.13851178  1.099062 
8  13.75  14.27491459  13.75  13.17571468  1.0992 
9  12.22222222  12.74822067  12.22222222  11.6488641  1.099357 
10  11  11.52670461  11  10.42717266  1.099532 
Rule Of 140
Taking a growth rate of r, the accurate value X is calculated as:
X = Ln 4 / Ln (1 + (r / 100))
Taking a growth rate of r, the accurate value Y is calculated as:
Y = Ln 4 / Ln (1  (r / 100))
Curiously enough this difference seems to equate to the Natural Logarithm of 4 (Ln 4 = 1.386294), which is where we get the Rule Of 140 in the first place. Again, a natural pattern is confirmed.
Ln 4 = X + Y
This can be written:
X = ABS( Y) + Ln 4
Growth Rate  Quadrupling Period  Quartering Period  X + Y  
Rule Of 140 
Accurate Value (X) 
Rule Of 140 
Accurate Value (Y) 

1  140  139.3214338  140  137.9351279  1.386306 
2  70  70.00557756  70  68.61923698  1.386341 
3  46.66666667  46.8995445  46.66666667  45.51314613  1.386398 
4  35  35.34597537  35  33.95949604  1.386479 
5  28  28.41339817  28  27.02681467  1.386583 
6  23.33333333  23.79132209  23.33333333  22.40461117  1.386711 
7  20  20.4895367  20  19.10267502  1.386862 
8  17.5  18.01293668  17.5  16.62590083  1.387036 
9  15.55555556  16.08646345  15.55555556  14.69922992  1.387234 
10  14  14.54508179  14  13.15762696  1.387455 
Conclusion
Although not as popular as the Rule Of 70, it was worth examining the Rule Of 110 and the Rule Of 140 to confirm the suspicion that the Natural Logarithm of a factor (in this case 3 and 4) appears to be roughly the difference between the time it takes a population to grow by that factor ( multiply by 3 or multiply by 4) and the time it takes to shrink by that factor (divide by 3 or divide by 4).
This brief investigation also confirm that shrinkage time is always the smaller of the two (the time to multiply and the time to divide).