Langara College - Department of Mathematics and Statistics

## Exponentials and Logarithms

The University of Saskatchewan's EMR site has brief tutorials and exercises on exponentials and logarithms and their applications.

The number e that is used as the base for "natural" logarithms can be defined either as a limit (which is closely related to the case of continuously compounded interest in finance); or graphically, as the unique base for which the exponential function crosses the y-axis with slope exactly one. Here's another version of the graphical approach. You can also explore both approaches with our own  Graph Explorer  (see its "Things to Do" link or our on-line lab for more details). The number e is aslo explained at The Natural Logarithmic Base, and is one of many numbers included in Favorite Mathematical Constants.

This "S.O.S. Calculus" help site starts with material on Exponentials and Logarithms and has been reviewed by Frank Choy
and this section on Exponentials and Logarithms  from "Karl's Calculus Tutor" also has been reviewed by Frank.

This worksheet from BC math teacher Doug Sly covers a number of properties of exponentials and logarithms, and includes an exercise exploring the relationship between ln(x) and the distribution of prime numbers.

This online tutorial includes both precalculus and calculus material.

Exponential Functions are also discussed (rather more abstractly) at the Analysis WebNotes site.
The same site introduces logarithmic functions in an exercise.

The sum rule for logs is illustrated graphically by Java applets from the IES group in Japan.
(Like many of the wonderful IES applets, these come with rather cryptic explanations. It's worth the effort to figure out what they are getting at, but some will probably find it puzzling)

#### Applications

One important application of exponential functions is to the growth of capital with compound interest. For an example consider the growth over 200 years of a bequest in Benjamin Franklin's Will (as discussed in a web site at the University of Pennsylvania).

Another application is population growth Population Growth (also at UPenn). (There was a nice specific example of this in Hawaii on Bee Parasitic Mite Syndrome - now lost, but I'm leaving this in in case they restore it).

An interesting application of logarithms is to the comparison of sensory and other intensities - examples include Stellar Magnitudes, the Richter Scale for earthquakes and the decibel scale for comparison of loudness of sounds.
There are also some applications discussed at the University of Saskatchewan's EMR site.
Various applications of mathematics have been collected by Eric Hiob of  B.C.I.T. (British Columbia Institute of Technology). Here are the ones involving exponentials and logarithms:
Belt Friction - An application of logarithms to mechanical technology
The RC circuit - An application of logarithms and exponents to electronics
How fast is this population growing? For how long can it continue?
An Application of Logarithms and Exponents to Environmental Health
An Application of Logarithms and Exponents to Occupational Health and Safety
Application of Logarithms and Exponentials to Food Technology
An Application of Logarithms and Exponents to Nuclear Medicine
Bode plots
You might also check our 'raw list' (of links provided without comment) to see if there are any
more examples there that we haven't yet included here.

If you have come across other good web-based illustrations of these and related concepts,