There are some students who, no matter what, can’t seem to comprehend what a logarithm (when treated like an operation) is doing. I see students that:
These mistakes indicate that “log” is being perceived as some sort of quantity to be manipulated, not as an operation. This may be due to the fact that “log” is the first time students are exposed to an operation that is represented as a word instead of as a symbol or other numerical notation. Texts apparently assume that this is a natural transition, not even worth mentioning, but it’s pretty clear that it is not as obvious as one might think.
To help students see what is going on, I’ve tried expressing other operations in a similar manner and drawing parallels. For example, take a look at roots and powers:
Logarithm does not have a symbol; our initial idea was to therefore rewrite exponentiation in terms of the “word operation" exp. We then explained that logarithms are the inverse of exponentiation, and that they undo each other, just like addition and subtraction, multiplication and division, and powers and roots.
This seems to have worked moderately well in terms of getting students to be able to evaluate and solve the log problems that they encounter on the STAR tests. However, I don’t think it’s really helped them to understand what a logarithm is, and their ability to apply the concept flexibly is quite limited.
I’m wondering now if going the other direction would have been better. Instead of rewriting exponentiation as a “word operation", we could have invented a symbolic representation for logarithms – say, a big L. (Not to be confused, of course, with the L formed by thumb and pointer finger, raised to the forehead!).
Inverse operations could then be modeled like this:
When I ask my students what “the third root of 8” means, they are pretty good about saying something like “what number to the third power gives you 8”.
When I ask them what “the log base 2 of 8” means, they rarely can say “2 to what power gives you 8”. I wonder if using a symbolic representation of logs will allow this meaning to be clearer. After all, when you think of a log in this way, it’s not really that much more confusing than a root.
I’d be interested in hearing any thoughts on this. Would a symbol for log be helpful? Confusing?