Understand that a logarithm represents the exponent to which the base must be raised in order to attain the input value; use this understanding to evaluate logarithmic expressions.
Use exponential and logarithmic forms to write equivalent statements about powers.
Understand the inverse relationship between how inputs and outputs change in exponential versus logarithmic functions.
Understand the inverse relationship between exponential and logarithmic functions of the same base, including the natural base, e.
Quick Lesson Plan
Students begin exploring logarithmic functions 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 x-value doubled, the y-value increased by 1. This is a helpful point to help students see that this mystery function has to do with the doubling function (an exponential function with base 2) 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.
Why were you not able to find an output for x=0?
Let’s add one more to the table. What if x=40? What do you think y would be? Can you give an estimate?
What do you notice about the x-values that produce negative outputs?
Can you think of an example where you might need to answer a question like Farah’s? How is this related to when we were talking about dog food?
This lesson ties together two topics in the AP Precalculus course framework: evaluating logarithmic expressions and understanding logarithmic functions as inverses. If you find that your students are new to logarithms or particularly rusty, you may wish to add an extra day before this lesson and do the Algebra 2 lesson that introduces logarithms.
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 of the logarithmic function 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?
One of the main goals of today’s lesson is for students to understand the inverse relationship between exponential functions and logarithmic functions. We have already seen how students can see this from the input-output pairs, but we also want students to see this from the way the variables change with respect to each other in each of the functions. This idea is drawn out in question 2 of the Check Your Understanding, especially part c.