Logarithmic Models and Logarithmic Scales - Maple Help

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Logarithmic Models and Logarithmic Scales

Logarithmic Models

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: $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$ It can be shown that this sum grows without bound, but it does so extremely slowly.

 Number of terms Sum $1000$ $7.485...$ $10000$ $9.787...$ $100000$ $12.090...$ $1000000$ $14.392...$

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.

References

 1 Human population estimates and projections: http://en.wikipedia.org/wiki/File:Growth.png.
 2 UN projections: http://en.wikipedia.org/wiki/World_population, accessed 2 December 2016.

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