##
A Simple Example --- Logarithms

This page illustrates two different approaches to learning
mathematics: the memorization approach which is widely
taught because it appears to be easier on teachers and
students, and the understanding based approach that's
ultimately much more effective. The example assumes that
you know what a
logarithm
is.

I know that logarithms with different bases are proportional
to each other, but I can never remember the factor that
connects them. But I can figure it out, and to this day I
figure it out anew every time I need to convert logarithms.

Suppose you are writing software for a computer that uses
the binary system and you require the logarithm with base 2.
You have a routine that computes the natural logarithm.

####
The Memorization Approach

In the first approach you'd remember, or find in a
collection of formulas, that two logarithms with base a and
b are related by the formula:

You recognize that you must set

This gives you the (correct) formula

####
The understanding based approach

I know that by the definition of the base 2 logarithm

I also know (since I understand some immediate consequences
of the
definition of logarithms
) that the logarithm of a power equals the exponent
multiplied with the logarithm of the base. Therefore, by
taking the natural logarithm on both sides of the preceding
equation I obtain:

Solving for the base 2 logarithm gives the same formula as
before:

But in the process of obtaining it I did not have to rely on
memory or reference works. It did require some mathematical
understanding in the vicinity of the fact I was interested
in, but if somehow I was unsure of that I could have
derived it similarly from simpler principles.

Fine print, your comments, more links, Peter Alfeld,
PA1UM

[17-Jun-1997]