Extraneous Solutions of Log Equations--A Graphical Representation

Last semester when I taught log equations, a curious student asked, "Why is it that we sometimes get extraneous solutions?"  A fantastic question that, at the time, I answered horribly.  I gave some mumbo-jumbo explanation about how extraneous solutions can appear when we combine logs. For example, for log(ab) to be defined, we need only ab>0, but that can be the case when a and b are either both positive or both negative.  If they're both negative, then they wouldn't satisfy the original expansion log(a) + log (b).

Not my finest moment as a math teacher.

This semester, I was determined to come up with a better explanation.  I gave my classes the following true/false question.  (I'm mean and didn't give them the option of "sometimes true.")

True or False:

logx+logx=logx22

The students said it's true (with no hesitation) and justified it with the Product Rule (Power Rule works, too).

Then I showed them this:

f(x)=logx+logx

f(x)=logx2

What the #$%@...?

"So, what's going on here?"  I asked innocently.

We had a good discussion about how the left graph is the same as the right if you restrict them to x>0 and how the right side would be defined for both positive and negative values for x since any real number squared is non-negative.  This (I hope) led to the powerful conclusion that the properties for logs have to be used with caution since logs are only defined for positive arguments.  This was stated when we first introduced the rules, but it's easy to forget.  Furthermore, when solving log equations, we don't know if the arguments are positive or not...so we have to use the properties and then come back and correct ourselves if need be.