A teacher approached me with the following question.
The function has a graph that looks like:
It makes sense that the function exists for all negative x values, because when you raise a negative number to the -2 power, you’re going to get a positive number. And you can take the natural log of a positive number.
Then the teacher said to consider the following function: , and the graph looks like:
Notice that you can’t input negative x values, because the domain of natural log doesn’t allow for it.
Here’s the question.
According to the log rules/properties, we know that:
So . But the graphs are different.
We went a little crazy trying to figure out what’s going on… For about 3 minutes, we were having a great conversation. But we quickly converged on the little text that accompanies the log rules in any textbook… and this text says that these rules work but are only valid for .
I kinda love this as an in-class exercise (I’ll probably forget this when I get to logarithms, but maybe posting it here will prevent me from forgetting it). Because it will force kids to (a) be confuzzled, (b) talk through ideas, (c) go back to the definition and qualifications for the log rules, and (d) see that these rules are indeed valid (we didn’t break math), but they are a bit more restrictive that we might have thought.
What I love is that isn’t actually an identity. But we are so used to using the rules blindly, robotically, that we never think about it. But for it to be a good mathematical statement, you need to qualify it! You need to say this is only an equivalence for . This was a good reminder for us.