Katharine Sparks WILLIAM CHRISMAN HIGH, INDEPENDENCE, MO
12th Grade Math : Unit #3 - Exponential and Logarithmic functions : Lesson #7

Change of Base Formula

Objective: SWBAT solve equations using only common logarithms and natural logarithms
Standards: HSF-LE.A.4 MP1 MP3 MP7
Subject(s): Math
60 minutes
1 Bell Work - 10 minutes

Today students will discover the change of base formula for logarithms. This formula is important in calculus. The derivative of logarithms with base b is converted to natural logarithms then the derivative is found. The definition of a logarithm of base b is defined using the change of base formula in Larson's Calculus, 8th ed. p. 361 (see also http://www.mathwords.com/c/change_of_base_formula.htm).

Students begin by working the problem on the board. Students have solved logarithms using inverses so determining how they would solve using a specified base is different. Students that are totally confused are asked to solve the problem using the process they know. By allowing students the opportunity to solve using what is understood may help students see how to do the problem when the base is not the same as the exponential function.

Once students have found a solution, I choose a student to show the solution using the method learned in a previous lesson and the method of using a common logarithm. After the answers are put up I comment that these answers do not look the same how we can verify they are the same. Students use their calculators to verify the 2 answers are equivalent.

 

2 Discovering the Change of Base Formula - 15 minutes

Once students have the answer to the Bell Work. I comment about how both answers have some commonalities. I ask students if they see some similarities. Most students notice that both answers have a log 12 as part of the result. I expect that some will notice that the denominator of the common logarithm answer has the base as the argument. Many will not be confident that this is meaningful information.

To help students explore this idea further, I will give them the Change of Base worksheet. Students work on this worksheet for about 10 minutes. As they work I make comments about how the answers to the first problem are similar to the bell work. I also note how the common logarithm is similar to the natural logarithm. I help students who are struggling, because I want them to focus on how the answers compare, not on how to solve.

The Change of Base worksheet two more basic exponential equations to solve. Most of my students will solve the problem using inverses, common logarithms and natural logarithms. After solving these 2 problems students rewrite a logarithmic expression using different bases and then write a general equation. Once  most students develop a conjecture on their own, I will have a student share his/her results for Problem 1. I will then ask a student to give the answers to Problem 3.

Students learning process
Student Led Inquiry

As students worked on the activity they quickly began seeing how to find the natural log equation without solving. This can be seen in the student examples (starting at page 3). The students knew that the only thing that would change would be the logarithm.  

When we began discussing the problems I could hear students say the base just goes to the bottom of a fraction. To make sure students we understanding, I had students work problems where they changed to different bases not just a common log an natural log. 

Pages 1 and 2 show a student that is not ready to look for patterns this student is wanting to learn a process and consisitently use the process for every problem. Before we ended the day the student had not finished the connection questions and just filled those in during the class discussion.  I will work with students at this level on a one-to-one bases. 

3 Practice and generalizing change of base - 10 minutes

For me to quickly assess the students' progress, I give students a couple of practice problems. I will ask students to work on  mini white boards. After a couple of minutes,  I will have the class show me their answers. As they do, I scan the room and note any mistakes or students that do not share.  I then have the students show each other the white boards. If it will help the class, I will put the answers on the board.

I now ask students to share their Change of Base Formula from Question 4 on the worksheet. Typically, my students will share several different ideas. To justify the correct answer, I will give students a problem with paremeters instead of numbers. The students work on the process and then share the result. After we agree on the Change of Base Formula, I ask students use the formula to rewrite a logarithmic equation as a common logarithm and natural logarithm.

Most years, my students want to know why we need to solve using both common logarithms and natural logarithms. I explain that general logarithms are defined as natural logarithms in calculus. I also explain that standardize tests use common and natural logarithms for answers and you may have a calculator that does not have the functionality to type in logarithms of different bases.  Since most of my students plan to take higher level math classes, this justification is enough incentive to move forward.

 

4 Closure - 5 minutes

To close this class I ask students to work on a problem. I chose this problem because it is more complicated than we have worked with today, but it is similar to problems worked in the Solving Exponential and Logarithms lesson. I want to see how students solve this problem.

There are two main methods I expect students to use. Some will solve using inverses and then rewrite the problem using the change of base formula. Other students will use natural logarithms as they solve. Some students want me to tell them which method to use. I explain that the problem can be solved using different processes, but everyone should get the same answer.

To prepare for the tomorrow's activity students are given Review Problems for station day worksheet. I ask students to work on the problems prior to class tomorrow as we will be discussing these problems in class.

Why the review sheet
Intervention and Extension

As I was working with students on solving, I realized that many students were confused with the items we had been discussing. I decided to make Review Problems for station day worksheet because many of the students were not doing any work outside of class. I wanted students to focus on some key ideas we had worked on in this unit. 

When the students received the worksheet they immediately began working on the problems. Some were upset that they did not know how to do the problems but this should indicate to the students that they may need some extra help.