Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

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