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Logarithmic Models and Logarithmic Scales
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Logarithmic Models
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A process is likely to satisfy a logarithmic model if how it evolves is inversely proportional to its current state. Informally, logarithmic processes exhibit rapid growth when they are small and slow growth when they are large.
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Logarithmic growth is very slow. Consider, for example, adding up the reciprocals of the integers:  It can be shown that this sum grows without bound, but it does so extremely slowly.
Exponential growth is very fast. Compound interest, growth of bacteria, runaway nuclear explosions all exhibit exponential growth.
Polynomial growth lies between these extremes. Area grows quadratically with respect to perimeter. Volume grows cubically.
If you plot a logarithmic function, a polynomial function, and an exponential function on the same graph, then for large enough values of the independent variable, the exponential function will outstrip the polynomial, and the polynomial will outstrip the logarithm.
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The Malthusian population model
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The Malthusian population model is essentially one of exponential growth punctuated by periodic catastrophic collapse. It is a reasonable model over the short term, but does not usually do a good job of modeling long term population growth.
A somewhat more realistic model is the logistic growth model. In this model, growth is initially exponential, but as resources become scarce, the growth rate steadily decreases as the total population approaches the maximum sustainable value, called the carrying capacity of the environment in which the population lives; the behavior is more logarithmic as the population approaches the carrying capacity. The equation for the logistic model is ![P(t) = P[0]*K/(P[0]+(K-P[0])*exp(-rt))](/support/helpjp/helpview.aspx?si=8421/file02292/math167.png) , where  , ![P[0]](/support/helpjp/helpview.aspx?si=8421/file02292/math173.png) ,  are carrying capacity, initial population, and growth rate, respectively, and  represents time.
Use the sliders below to choose the carrying capacity,  , the initial population, ![P[0]](/support/helpjp/helpview.aspx?si=8421/file02292/math189.png) , and the rate of growth,  , and observe how the population growth model varies over ten time periods.
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Carrying Capacity,  :
Initial Population, ![P[0]](/support/helpjp/helpview.aspx?si=8421/file02292/math235.png) :
Rate of growth,  :
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The graph at the right shows estimated total human population for the past 2000 years, and projected forward to 2060. The vertical scale is logarithmic, and the curve still generally curves upward. This suggests that in fact human population growth is super-exponential, faster than exponential.
However, current United Nations estimates suggest that we are rapidly approaching the carrying capacity of our planet and that the population growth rate should start seeing a significant decline in the next hundred years.
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Human population from 1 A.D. through 2060 A.D. (estimates and projections)
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