Today we are going to be spending some time solving equations that involve exponential expressions and logarithms. Typically these equations are difficult for students because exponentials and logarithms are difficult for students; it is usually not the algebra that gets them. However, sometimes my students will just look at these equations and instantly freeze up because there is a logarithm in it, and that is what I want to avoid.
The purpose of this lesson is to show two equations that are analogous to equations that they can easily solve. In the lesson, I try to make the connection perfectly clear. I think that this helps to reduce the "intimidation factor". My goal is that if students encounter an equation that they do not instantly know how to solve, they will make a connection to a known strategy that they will be able to use. MP7 (Look for and make use of structure) is vitally important for today's lesson. Students will use the structure of these analogous equations to solve problems that they might not have been able to do otherwise.
To launch, tell students that we are going to spend the day solving equations. I do not want to tell them much more than that because I want them to discover that the equations on each side of the worksheet are related. Give students the task worksheet and have them work in their groups for about 25 minutes. Walk around the room and monitor their progress and take notes on what you are observing, because you will probably want to discuss some things you notice (both positives and negatives) as a class.
On the front side, I am sure that students will have few problems with solving questions 1 and 2 by factoring. When they get to 3 – 5 they may slow down. Ask students to notice what is consistent among all five of the equations. Then ask them how they solved questions 1 and 2. This comparison will get them thinking about using factoring to solve, even though these equations are not as simple.
For question 5, students may need a reminder that e2x = (ex)2. Keep an eye out for the need to review some exponent rules. Also, students may produce an equation like ex = -2 once they set their factor equal to zero. If they try to do ln(-2) on their calculator, it will produce an error message. Students may think they have done something wrong. Tell them to discuss this error with their group. But, let them know that we will eventually address the issue as a class.
The back side of the worksheet will likely be a little more difficult. The progression from the first question to the last question is much more rigorous than on the front side. If the students have recognized that they can use the same techniques throughout the front side of the worksheet, they will probably want to do the same for the back side. The main strategy on the back side is to rewrite the exponential equations in logarithmic form and use that to solve.
Start by asking students what they noticed about the problems on the front side of the worksheet. They will probably talk about how the structure of the problems is the same. Ask them to be very specific so that everyone realizes that all are of the form b2 – 4b – 12 = 0 where b stood for very different things. Ask them how they used their past experiences with quadratic equations to solve these problems. The nice thing is that the zero product property works for anything (variables, exponential expressions, logarithms, tangents, cosines, etc.) and this set of analogous problems will allow students to see that. After talking about the similarities, ask a student to show an example of problems 3 – 5 and make sure that your students understand how to solve these equations. For question 5, make sure you talk about the fact that there is only one solution and discuss why that is. Students may not immediately notice that ln(-2) is impossible to compute, but writing this as ex = -2 might make it more clear that this cannot be solved. Generalize this to say that you can never take the log of zero or a negative number.
For the back side, again ask what the problems all have in common. Get students to see that they are all in exponential form and must all be solved using logarithms. Go through the first problem – students should have no trouble solving this – and then make sure you solve the rest of the problems in the exact same way. In the video below, I discuss the logarithmic forms of these equations and how they are related.
Questions 8 and 9 can get a bit tricky after putting the equation in logarithmic form. They are going to have to move the exponents to coefficients (one of our logarithm properties) and the algebra is going to get a bit tricky. One approach is when they have something like log25x, students can evaluate log25 and write a decimal – that will simplify the algebra tremendously. Then the problem becomes a simple exercise of isolating your variable.
To finish up the discussion ask students to think about the following questions and discuss among their group. These questions are a good way to summarize the work we did today and to give them closure on the topic of solving equations with logarithms and exponential expressions.
1. Why do you think I had you do questions 1 – 3 instead of just giving you questions 4 and 5 to solve?
2. What made questions 4, 5, and 9 more difficult than the other problems on the page? What made them difficult?
3. What strategy can you use if you encounter an exponential or logarithmic equation that you do not know how to solve?
I want students to know my teaching strategy behind this lesson because it will solidify the fact that these equations are very similar in structure to problems that they do know how to solve, but they are made slightly more difficult by the presence of exponential expressions and logarithms. The main idea is if they get another equation they are stuck on, they can simply treat the exponential expression or logarithms as a variable and see where it gets them.