Do you have those students in class who sincerely want to understand why? And this question of why came from them playing around with a few things and noticing a pattern? Have you ever been asked one of these questions in the middle of class when you are teaching something such that this new additional topic would completely lose your students?
What do you do in this case?
Scenario: "Miss Rudolph!!!!!!!!!!!! MISSSSSSS RUDOLPH!!!!!! Can you explain to me why ln(4)/ln(5) = log(4)/log(5)????????"
Me: Wait, where did this come from? (We were in the middle of another problem so I was caught off guard) ..... After a few moments....... Can I look into this a little and get back to you?
Child: But, but, but, but.... fiiiiiiine
So I tried googling it a little, because that's what everyone always does, right? Because of course I don't remember all the proofs that I did in college.... good times..... But I couldn't find it.
Now, let it be known, I put all of maybe 5 minutes into googling when I had other students come in that afternoon so I decided to provide the child with the additional assignment.
The next day in class... "Miss Rudolph, did you find out why?"
Me: No, I did not. So I give you this challenge, you prove it to me. You find out why, and I will give you points.
This child looooooooves points. So needless to say, the game was on.
The following morning:
Child (walks straight into my room): Okay, give me the points now. I've done it!!
The child proved ln(a)/ln(b) = log(a)/log(b) and let me tell you, wow do I feel silly for not having done it myself.
Let's think of it using the Change of Base Property!
ln(a) = Log(a)/Log(e) and ln(b) = Log(b)/Log(e)
Well, do a little division magic and poof! You get log(a)/log(b)
I felt like a very silly math major.... but the child got his points.
Challenge your kids, they may or may not surprise you. This lead into a discussion about possibly looking into being a math major. I informed the child that the proof and process that he went through by using properties was exactly what I spent 4 years doing.