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.
"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.
Functions are also discussed (rather more abstractly) at the Analysis
The same site introduces logarithmic functions in an exercise.
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)
If you have come across other good web-based illustrations of
these and related concepts,
please do let us know and we will add them here.
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