Derivatives of Exponentials and Logs with Desmos

Here's a Desmos activity I typed up for my calc kids to find the the derivatives of exponential and logarithmic functions.  Sadly...the class set of laptops would not connect to the domain because they hadn't been used all summer, so we did this together as a class, which was not what I wanted, but what can you do?  Instead of having the kids click pause on their own screens, I had them yell "STOP" at me...so it was still entertaining.

All this to say, I don't know if this is good or not since I haven't gotten to test it out on students yet, but here it is.  Use/modify if you'd like!

Derivatives of Exp and Logs Desmos

In the "notice/wonder" section of f(x)=a^x, one student said he noticed that the derivative was proportional to the given function.  This made me a very proud momma and was a perfect segue into finding the derivative when a is different from e.  (We explored f(x)=2^x, f'(x), and g(x)=lna(2^x), and found that a=2).

Desmos also sent me this great online activity.

Solving Exponential and Log Equations Flow Chart

I guess I've been kinda into flow charts this year; I created another one for solving exponential and log equations.  The idea is that kids start with the top box, if they can't complete that task, then they go on to the next box.  We put examples in each box.  I used this for both Algebra II and Pre-Calc this year.

It's not flawless, because mathematics requires more creativity than a flow chart can provide.  But it gets the basic ideas across.

Solving Exponential and Log Equations Flow Chart

Exponential/Log Function Review Day + Napier!

I've written before about how review days are a continual source of stress for me.  To review or not to review? that is the question.  And if to review, how do you make it interesting and beneficial for the upcoming test?  I have no profound answers yet.  But, I am putting a lot more time and effort into my review days now (mostly because I've taught College Algebra enough times so that I have the extra time to do that).

That said, I've been very much looking forward to this review day for quite some time.  The unit has been on exponential and logarithmic functions.  When I talk to my husband about this unit (who patiently listens to all my teacher talk--I found a really good man, let me tell you) he always reminds me, "It would be much less scary if it weren't called a LOGARITHM."

And he's completely correct.

So, I've been hyping up this review day nearly all unit:  "We'll talk about who invented logarithms, why in the world he did so (just to make your lives miserable?), and why they're called what they're called."

Here are the slides that took us through a short history of logs.  The SMART notes didn't convert perfectly to PowerPoint, but email me if you want the SMART notes as well.

What Smarty-Pants Invented Logarithms and Why the #@&% Did He Do So?!

View more PowerPoint from Rebecka Peterson

I gave the students a sheet that corresponded with the notes so they could follow along.
After we went through John Napier's method of multiplying two numbers we worked the same multiplication on slide rules!  A few of my colleagues were kind enough (and...ahem...old enough) to loan me enough slide rules so that nearly every student could have his/her own for the day.
Next, as review for the test and as further proof of how quickly exponential functions grow, I had the class break into four groups and choose one of the following problems below.  I got the first problem from Ethan Siegel's blog.
Unit 4 Review Problems


We ended with a short wrap-up of the big ideas of this unit.

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.

Log to Exponential Form & My Two Favorite Words

I don't know where I first I heard this, but I've had such success with it in both my teaching and my tutoring that I wanted to share it.  When students are converting from logarithmic to exponential form (or vice versa) there are two words that have become my new best friends.

Ready?

Basement and base.

For the most part, students seem to know exactly where to go when switching from one form to the other as long as they start out right.  How do I ensure I start out right?  I just remember:  "What's in the basement becomes the base."  (Or vice versa)

Does this solve all my problems?  No.  Does it guarantee that students fully understand what a log function is?  Absolutely not.  But, I want my students to be able to covert from log form to exponential form like it's nothing.  And this helps that process some.

Maybe this is widely known already.  But I still heart it.