Solving Exponential Equations

Lesson Details

Grade Level: 12th Grade
Subject Area: Algebra II – Solving Exponential Equations
Time Duration: 45 minutes
Curriculum Standard: CA Common Core State Standards:

• HSN-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.
• HSA-SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
• HSA-REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).

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Learning Objectives

By the end of this lesson, students will:

1. Understand the process of solving exponential equations algebraically.
2. Apply the properties of exponents and logarithms to simplify and solve equations.
3. Interpret solutions in the context of problem-solving.

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Materials Needed

• Whiteboard/Markers
• Student Notebooks
• Graphing Calculators
• Printable Practice Problems (provided in the “Guided Practice” section)
• Timer/Stopwatch

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Lesson Overview

Engage (5 minutes)

1. Begin the class by posing a real-world question to capture interest:
   “Imagine a phone company advertising that their customer base grows exponentially—doubling every month. If they start with 500 customers, how long will it take them to reach 8,000 customers?”

2. Allow students 2–3 minutes to brainstorm ideas. Discuss briefly how exponential equations model such scenarios and how solving them is essential in fields like science, economics, and technology.

3. Write on the board: “Today, we’ll learn how to solve exponential equations algebraically.”

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Explain (15 minutes)

1. Recap of Prior Knowledge (2 minutes):

   • Quickly review the properties of exponents:
     • (a^m \cdot a^n = a^{m+n})
     • ((a^m)^n = a^{m \cdot n})
     • (a^{-n} = \frac{1}{a^n})

2. Definition of Exponential Equations (2 minutes):
   Write on the board:
   “An exponential equation is an equation in which a variable appears in the exponent.”
   Example: Solve for x in (2^{x + 3} = 16).

3. Teaching Methods (11 minutes):

   • Case 1: Matching Bases:

     • Rule: Rewrite both sides of the equation with the same base.
     • Example: Solve (3^{x + 1} = 81).
     • Solution: Rewrite 81 as (3^4), so (3^{x + 1} = 3^4 \Rightarrow x + 1 = 4 \Rightarrow x = 3.)
     • Ask students to solve (5^{x} = 125) on their own.

   • Case 2: Using Logarithms:

     • Rule: Use logarithms to isolate the variable if matching bases are not possible.
     • Example: Solve (2^x = 10.)
     • Solution: Take the natural logarithm (ln) of both sides:
       [ \ln(2^x) = \ln(10) \Rightarrow x \cdot \ln(2) = \ln(10) \Rightarrow x = \frac{\ln(10)}{\ln(2)}. ]
     • Plug this into a calculator to find (x \approx 3.32.)
     • Discuss how logarithms are powerful tools in solving exponential equations.

   • Case 3: Real-World Applications:

     • Example: Return to the initial phone company scenario: (500 \cdot 2^t = 8000.)
     • Solve step-by-step:
       [ 2^t = \frac{8000}{500} \Rightarrow 2^t = 16 \Rightarrow t = \log_2(16) = 4. ]
     • Conclude it will take 4 months to reach 8,000 customers.

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Explore (10 minutes)

1. Guided Practice (5 minutes):
   Distribute a worksheet containing the following problems:

   • (4^{x - 2} = 64)
   • (10^{2x + 1} = 0.001)
   • (3 \cdot 5^{2x} = 375)

   Walk around the room, checking student work and answering questions. Emphasise breaking down the steps logically and neatly.

2. Group Activity (5 minutes):
   Pair up students and provide each group with a harder problem:

   • (7^{x + 1} - 3 = 24.)
     Ask them to solve and explain their reasoning to the class.

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Elaborate (5 minutes)

• Challenge students to connect exponential equations to other subjects, such as:
  • Biology: Exponential growth of bacterial populations.
  • Physics: Radioactive decay.
  • Economics: Compound interest formulas.
• Pose a reflective question: “What happens if logarithms didn’t exist? How would this affect our ability to solve real-world problems involving exponential equations?”

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Evaluate (5 minutes)

• Review the objectives:

  1. How do we solve exponential equations with matching bases?
  2. How do logarithms help when bases don’t match?
  3. What are some real-world applications of exponential equations?

• Homework Assignment: Assign the following problems to solidify student understanding:

  • (2^{x + 4} = 64)
  • (e^{2x} = 20) (Use the natural logarithm function.)
  • Research real-world uses for exponential growth or decay models. Bring one example to discuss in the next class.

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Reflection for Teachers

• Note if students grasped both matching base techniques and logarithmic methods.
• Assess engagement through student participation in guided and group activities.
• Look for evidence of high-level thinking during the exploratory group discussion.