Logarithmic Functions (Lesson 3.4)
Unit 3  Day 5
Learning Objectives

Understand that logarithms tell you how many times the base must get multiplied (the exponent) to arrive at a certain value (the input)

Convert equations from exponential form to logarithmic form and vice versa to solve simple exponential and logarithmic equations.

Use the natural log to find the inverse of e and vice versa.
Quick Lesson Plan
Experience First
Students begin exploring logarithmic function by thinking about a mystery function and analyzing its inputs and outputs. It may take a while for students to find a pattern. Students initially guess the square root function since 16 goes to 4 and 4 goes to 2, but they quickly realize that since 32 goes to 5, and 8 goes to 3 this rule doesn’t work. Most students were able to figure out that 64 goes to 6 because they noticed that every time the xvalue doubled, the yvalue increased by 1. This is a helpful point to help students see that this mystery function has to do with the doubling function (base 2 exponential function) and specifically that 2^y=x. Once students find this, they are able to deal with rational inputs and negative outputs.
Students will realize that some values in the table are impossible to find (negative values and 0) and some values are simply impossible to find by hand such as 2^y=3. As students discuss this in their groups, ask them to explain why they were not able to find certain outputs. Most students will be able to articulate that 0 and negative numbers won’t work because when you raise 2 to the 0 power you get 1, not 0, and raising 2 to the negative power gives a fraction, not a negative number.
Formalize Later
The debrief portion will formalize the idea of a logarithm which may or may not have come up in students’ discussions, depending on their Algebra 2 background. Show students that logarithms are simply an alternate way of presenting the information in the table. Remind students that the functions they were familiar with from previous lessons looked like 2^x=y and then ask them what is different about this mystery function. Students should be able to articulate that the inputs and outputs have switched, and arrive at the idea of the inverse function. When discussing the domain of the logarithmic function, you can discuss the range of the exponential function being greater than or equal to 0 to help support their conjectures about which values are impossible to find.
Continue to emphasize to students that the logarithm gives (outputs/represents) the exponent. Although the language around logarithms can be clunky, we have found some success with referring to the input as the “desired end result” and the output as the exponent because it reminds students that the input is the output of the exponential function. Stay away from phrases like “the input is the big number” as it can muddle the mathematical ideas. It is very helpful to think about logarithms and exponentials as cause and effect, respectively. Exponentials answer the question: what is the effect? Logarithms answer the question: what got you there?