# Inverse Functions and Logarithms

After the Zombie exponential problem, several students wanted to know WHAT was this mysterious LOG thing and especially why it worked?  Enter, inverse functions.

I made an Inverse Functions guided worksheet for my students.  It is self-directed and has them use two different colors for the graphs.  They can get through the entire sheet without much help, and make all sorts of great discoveries on their own.  I blew it in the first class I taught by trying to go over the sheet with them way too soon after handing it out.  I wanted to explain a couple of things before they got too far.  DON’T DO THAT.  Give them ample time to read it, graph things, make small mistakes, and discover.  It makes a huge difference not only in understanding, but in their engagement and attitude.  They really don’t want to hear me talk.  Instead, play music in the background, walk around and see their work, even answer a FEW of their questions.  It is so good for them to work on their own!

After we went over the worksheet together, we played with the awesome inverse graph created by Desmos that I modified with the equation y=2^x.  The kids can move the slider and it shows the point and its inverse on each graph!

I created the graphs with an amazing program that Mrs.Davis found called GraphFree.

# Creating a Need for Logarithms with Zombies

Every since I have taught math I have noticed that students are not big fans of logarithms.  I think it may be because when they are in Algebra 2, logs seem so different from everything else they have done so far in math.  To solve a quadratic equation, you apply the square root; the inverse of x squared is the square root of x.  Equations they can easily solve like that make sense to them.  However, to easily solve most exponential equations, you need to apply the LOG.  Log doesn’t look like the exponential function at all, and I think that is where the confusion begins.

So this year, I started with a simple Zombie Word Problem (from Wendy).  The crisis?  Students were turning into Zombies at an exponential rate!  How many hours until our entire student body (1002 students) were ALL Zombies?  The catch?  I had not taught logarithms yet.  I had never even said the word LOG in class.  We had only done exponential functions and equations.  My students were knew how to solve equations like 2^x=8.  But, when given 3^x=1002, they were stumped.  3^6 was too small, and 3^7 was too large.  I insisted they could find a solution, and to try things.  They did amazing work.

After determining that we would all perish between 6 and 7 hours, they started the slow process of plugging decimals.  Every minute counts!  They also went to Desmos.  Some kids even spent quite a while plugging in numbers until they got the EXACT answer.  But many of my students remarked, “Is this how we are supposed to do this, just by guessing?”  I explained that they were not merely guessing.  It was a method that worked, derived from an original estimate.      But, they said, this takes too long.  “THERE HAS GOT TO BE AN EASIER WAY!”  I finally relented (wink, wink) and showed them the LOG feature.  Some of my students had heard of logarithms, and already had a negative opinion of them.  But today they were all happy to meet logs because it was such an easier method.  And now they were very curious to know what the heck this log thing was and WHY it worked.  Boom.

Today, awesome hook for instant interest in logs; tomorrow, inverse functions to show why logs work.

Zombie Word Problem – Smart Notebook, PDF