# Solving Equations Involving Exponents

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## Objective

SWBAT solve exponential equations by reasoning about exponent values and by graphing functions.

#### Big Idea

Students will use standard A.REI.11 to solve exponential equations without using logarithms.

## Launch

5 minutes

Any time a mathematical property/theorem has a restriction, it is important to have students think about why the restriction exists.  By doing so, we help students make sense of properties at a deeper level (MP3).  In order to accomplish this in today's lesson, I begin class by asking students to read and make sense out of the solve_exponential_equations_launch individually. After about 2 minutes, I ask my students to do a Think-Pair-Share to work together to put the information in the slide into their own words.  I also encourage my students to think about the question at the bottom of the slide regarding the base not being equal to 1.

To conclude the Think-Pair-Share, I will have students share out their understanding of the property defined on the slide.  As they share, I prompt them to come up with additional examples of this property, similar to the one given on the slide. I will keep track of these examples on the board so that students see many instances of this property.

## Direct Instruction

15 minutes

After the launch we will continuing working from the Solve Exponential_Equations_Direct  presentation.

Slide 1

I plan to have students turn-and-talk with a partner about how to solve the first equation (MP3). Some students may observe that the goal is to get the base/exponent portion of the equation isolated on one side.  This can be done in the first equation by multiplying both sides of the equation by 1/3 (or dividing each side by 3).  I plan to remind students that this "move" will lead to another true equation. At this point, all that is left is to figure out the power that represents the value of x.  Since this example is designed to result in a whole number answer, I expect some students may see right away that 2^4 is 16.  Others may make a list of powers of 2.

Although more complex, the second equation is similar to the first. I am interested in seeing what my students do as a first step. Then, I hope that my students see that both the numerator and denominator of the fraction must be raised to the same power. Again, some students may not see right away that (5/3)^3 is 125/27. In this case, I may proactive and suggest that students make a list of powers of 5/3.

Slide 2

When we turn our attention to Slide 2, I plan to have my students do a think-pair-share. I expect the first part to be relatively straightforward for students. I ask them to make a prediction in order to forestall the answer, "It's impossible."

Making a prediction of the value of x with some accuracy requires students to reason about the powers of 2 and interpolate (MP2).  For example, a student may say:

There is no power of 2 that will equal exactly 10.  2^3=8 and 2^4=16 so we know the power must be in between 3 and 4.  Since the value of 10 is closer to 8 than it is to 16, we could guess that the answer would be around 3.3.

This is a fairly comprehensive answer.

Most likely, you will be able to question your students in order to guide them towards this type of understanding.  This gets students thinking about the fact that not all exponents have to be whole numbers.  With this in mind, they can attempt the next question.

Slide 3

I ask students to begin to work on this problem with a partner. I encourage them to work together to set up an equation that could be used to solve this problem (MP1). I intend for my students to base their equation on the exponential growth model we have studied inprevious lessons.  Once students produce their equations, I will ask them to estimate values for x and then try to use guess and check to determine a solution.  Once students have had some time to think, I plan to inform them that there is a way to solve this equation by graphing (if they have not thought of this themselves).

I will begin by reminding students that two sides of an equation can be thought of as different functions (A.REI.11). If we take this approach, we can then graph both functions and determine the point of intersection by visual inspection. In this case, the two functions can be written as f(x)=100,000 and g(x)=79,918(1.06)^x.  After discussing this, I will ask my students to graph the functions on their graphing calculators (or using Desmos graphing software -- see solving_exponential_equations_solution).

## Practice

15 minutes

Using Solve_Exponential_Equations_Practice, students will be able to practice what they have learned so far. For this activity, I will ask my students to work with a partner.  I will encourage them to focus on being precise and supporting each other with respect to this mathematical practice (MP6). I will also require my students to check each answer by testing their answer in the original equation. This practice will be especially valuable for equations that have non-integer answers.

Questions 1 to 6 require students to solve an equation by recognizing the power that would make the equation true.  If they are stumped, students can also make a list of powers until they reach the correct result. For example, for 3^x=81 students could make a list of powers of three 3, 9, 27, 81 and determine that the correct value for x is 4 (x = 4).

The final three questions require students to reason about the value of the exponent before graphing the two equations and finding the point of intersection.  I make sure to clearly state these instructions to my students.  I think that it is important for them to make a prediction about the value of the exponent by reasoning before working on an exact answer.  Mathematically proficient students reason about the solution in many different ways to ensure it is the correct one (MP6).

## Close

5 minutes

This Ticket out the Door is an application problem that students will have to read and make sense of (MP1).  I plan to examine the student work closely to determine which parts of the question my students are able to complete:

• writing the equation
• breaking the equation into two functions
• graphing the functions
• finding the point of intersection

Students can estimate the value by raising 1.08 to various powers to determine how quickly it is growing and how long it will take to reach a value of 2.  They can then graph the two functions to find the exact number of years (9.006 years).