Note to Teachers
This text is not written for you. With the exception of parts of chapters 5,
6, and 7 and Appendix A, I assume that you already understand all the ideas
presented. This is a book written for students who do not understand
logarithms even if they can apply the rules and get correct answers.
However, it would greatly gratify me if a teacher were to tell me that he or
she enjoyed my organization and presentation.
I am a high school math teacher, not a mathematician. As such, I live and
work in a world where sequence and progression of concepts leading to key
ideas, along with pacing, “anticipatory sets,” evolution and organization of
ideas, reinforcement, examples and counterexamples, patterns, visuals,
repeated threading and spiraling of concepts, and, especially, repetition,
repetition, and repetition are all more important than rigor. It has always
seemed ironic that authors and teachers, so knowledgeable about mathematical
sequences, could be so insensitive and clumsy about the sequencing of
curriculum … how they could be so knowledgeable about continuity of
functions but so discontinuous in their writing. There are plenty of
materials available on teaching logarithms that are mathematically rigorous.
I believe that “rigor before readiness” is counter-productive for all but
the most gifted students.
As such, I present many, many examples to help the student to see patterns
and only then do I present the abstraction which will allow for
generalization to all cases. Induction is a powerful teaching tool. Because
of economy imposed by the publisher or perhaps because the material is so
“obvious” to the authors most textbooks present the abstraction
(generalization) first with little attempt to develop the rationale behind
it or to connect the material to previous material such as the Algebra I
Laws of Exponents or the history of logarithms. Those texts then proceed
hurriedly to applying the abstraction to specific situations. I believe that
the best way to introduce a new idea is to somehow relate it to previous
ideas the student has been using for some time. Using this approach, new
concepts are an extension of previous ideas … a logical progression.
Logarithms are a way to apply many of the laws of exponents taught in
Algebra I. It is important that the students understand that!! I also
believe in introducing an idea in one chapter and revisiting that idea
repeatedly in different ways throughout the book.
The materials presented here are usually spread over two years of math
instruction: precalculus and calculus. Doing so, however, separates ideas
and examples that are helpful in the synthesis that leads to a deeper
understanding of logarithms. For example, most high school text books seem
to shy away from a meaningful discussion of why scientists and other
professionals prefer to work with base e, the natural log, rather than the
more intuitive common base, base 10. They do so because the pre-calculus
student has not yet been exposed to the ideas that are necessary to justify
the use of base e. If the goal is “rigor” then indeed many ideas associated
with e must be postponed until calculus. But if your goal is to create
familiarity with logarithms and appreciation of the number e, I do not
believe that all that rigor is required. I have tried to bring all those
ideas down to the pre-calculus level. I hope that I have done so.
My approach, however, has been done at the expense of rigor. If I get
consigned to one of the levels of Dante’s Inferno because of my
transgression it will be worth it if I am able to help young students past
what, for me, was an unnecessarily difficult multiyear journey. When I did
make an attempt at “rigor,” I chose the formal two column proof over the
abbreviated paragraph proof.
I see three different audiences for this text: 1.) students who have never
worked with logarithms before, 2.) those students in calculus or science who
did not manage to master logarithms during their algebra/pre-calculus
instruction, and 3.) summer reading for students preparing for calculus. The
former students will need to receive instruction, but the second and third
group of students, if sufficiently motivated, should be able to read these
materials on their own with little or no help. There are questions at the
end of each chapter to use to evaluate student understanding. Heavy emphasis
is placed upon practicing estimation skills!!!