Today students begin solving logarithmic and exponential equations. I develop solving equations with these functions by discussing how the process is just like solving any algebraic equation.
Students come into class with 3 algebraic problems to solve. These problems demonstrate the main methods used to solve logarithmic and exponential functions.
The first problem is a 2-step linear equation. Students all know how to solve this so students quickly gain confidence that this lesson is not too hard.
The second equation is a quadratic. Students must rearrange the equation and then remember how to solve a quadratic. This problem is easily solved by factoring. This problem gives students the most problem. Many students are not proficient with solving equations and try to isolate the x^2 instead of setting the equation to zero. I identify students who struggle so that I can assist them with solving quadratics later.
The last problem show students demonstrates how you can solve when you have f(a)=f(b) Since both sides have a square root the inside functions must be equal. This is a useful property. This problem also reminds students that it is necessary to check their solution to verify the value found fits the domain for each function in the equation. The answer here is -12 which will not work for this equation. We briefly talk about how setting the insides equal has change the type of functions we are solving. When the type of function changes the domain may change and that is why we get extraneous solutions.
I now give students a pair of exponential equations to solve. I will give my students a couple of minutes to work on the first problem. I expect that some will be unsure how to finish the problem. If so, I will ask students how they would solve the problem if the problem was x^2=55. Most students will say square root. If so, I'll ask, "Why square root?" Someone usually identifies this as the inverse operation. At this point I (or a student) will say, "What is the inverse of e^x?" Once students remember the inverse is ln x we take the inverse of both sides and find the answer.
Additional questions my students often ask:
After we discuss the first problem, the students move to the next problem. Students will solve this problem and share their process with the class. I find that my students usually solve exponential equations with relative ease. Some students will try to divide by the base. When this happens I try to show the student what they really did. We will then discuss how you need to use the inverse.
At this point I have not discussed the change of base formula or how we need to solve using common logarithms or natural logarithms. As I progress through the unit I will use solving to show the change of base formula.
Today we are again working from exponential to lograithmic, so I will now give students several logarithmic equations to solve. We will work one problem at a time. As I move around I think out loud to help students think about the problem. I say "this has logarithms but does this problem remind me of one in the bell work?" Some students will realize it is like the third problem. These are usually the first to find the answer.
After 2-3 minutes, I bring the class back together. I have a student share their process. I usually have 2 different processes used. One is to see that both logarithms are the same so the arguments have to be equal. Other students take the inverse of each side. We discuss how both methods are correct ways to begin the problem and both yield the same answer. After a solution is found I ask the students how we know that the answer is not extraneous like the Bell Work answer. This is a time to review the domain of a logarithm. I remind students that the argument has to be positive. We check the answer and move on to the second problem.
As we start to work on the second problem, I'll ask how this problem is different from the first. I expect my students to observe that we are working with a natural logarithm and only one side has a logarithmic expression. I let students work on solving and we share the answer. I accept both the exact value and the calculator answer. Many times students will ask which answer I want. I tell the students that directions for the problem will explain. I also let students know that I do not usually require using the calculator because I am looking at their process of problem solving.
We move on and work each problem. The last problem is the most difficult for the students. Some students want to just remove the logarithm off the terms like they did with the first problem. I remind them that this was possible because we have one term on each side of the equation. Here we have 2 terms on the right. Students then realize they need to use a property to condense the right side. Once this is done students are able to solve.
Now that students have practiced solving I ask each group to develop a set of steps or questions they could follow in solving exponential and logarithmic functions. I have the students put their steps on white boards and share them with the class. Once all groups have shared we look for common steps or questions to develop a class process for solving equations.
When students write out their own process for solving they establish their own reasoning skills. Most groups have similar steps but some groups think of important ideas that others may forget. If a student is not able to write their own process seeing how other students solve gives them a process to use.
Students usually say they need only one logarithm on each side of an equation. You have to get the exponent term alone. You add/subtract first. You should check the answer. As students become more proficient they will add to this list.