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Wednesday, November 02, 2005

Logs

Why are logarithms so difficult? Algebra students who are consistently competent in other topics often stumble when they get to logs. Sure, they can memorize an algorithm for switching from exponential form to logarithmic form and vice versa, and most of them can use logs to solve an exponential equation as long as a straightforward algorithm can be applied without too much thinking.

But...vary the context a little, and bewilderment sets in. Even the calculator isn’t much help, since it won’t work with bases other than 10 (or e) except by means of a generally incomprehensible formula — yet another thing to be memorized rather than understood. So, we try to provide contexts involving exponential growth and decay, where logs provide the missing link: we know that we always need the inverse when we know the output but not the input, and we know that a logarithmic function is the inverse of the corresponding exponential function, so all is clear.

Not.

Perhaps the nub of the problem is that a log is really both a noun and a verb. When we write 2 = log3(9), we are simultaneously saying two different things: 2 is a log, so “log” must be a noun; and we are also applying the process of “taking the log” of 9 — without ever writing the word “taking”! — so “log” is now functioning like a verb.

I really think that this confusion is a major source of students’ difficulties, but I don’t understand why the same confusion doesn’t crop up in the case of powers, which also play the same dual role. In the example in the previous paragraph, we could write the equivalent exponential sentence, 9 = 32, and almost nobody seems to be confused about the fact that we have both the process of squaring and the resulting perfect square. Maybe it’s because we only have numbers in this case — no unfamiliar words like “logarithm.”

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