The standard explanation is this, “a logarithm is an exponent”.
This is true. But I’m not sure it’s particularly helpful for a student who is struggling. I have been burning a lot of mental calories over the past few versions of College Algebra trying to come up with ways into logarithms that will have more explanatory power and be more intuitively inviting to my students.
I realize the quest may be quixotic.
And I understand that this is well-trodden ground.
But consider this equation:
What if instead, we thought about it this way: How many factors of 2 are in 9?
More than 3, fewer than 4.
One has no factors of 2 in it, so:
because the expression on the left asks how many factors of 2 are in AB while the one on the right asks how many are in A and how many are in B, then adds them. Some of the 2s are in A, the rest are in B.
The language is problematic, I know. The answer to the question How many factors of 2 are there? is properly “2”.
But we use the language of factor in our exponential work, so it may not be too problematic in this context (and few College Algebra students are number theorists).
And of course I’m not naive enough to expect that this will solve all of our logarithm difficulties. But I’ve got something to work with.
Thoughts and critiques?