Solving Exponential Equations

Overview

Year group: Year 10
Duration: 60 minutes
Class size: 1 student
Unit: Mastering Algebraic Concepts (Lesson 17 of 20)
National Curriculum references:

• Key stage 4 (Years 10-11)
• Mathematics programmes of study: Number – Laws of indices
• Algebra – Use and manipulate algebraic expressions, equations and formulae
• From the National Curriculum for England (2014) –
  • Use and interpret algebraic notation, including: the form ax^n for positive integer n
  • Understand and use the laws of indices to simplify and manipulate expressions
  • Solve equations involving powers and roots

Learning Objectives

By the end of the lesson, the student will:

• Recall and apply the laws of indices when solving equations
• Solve simple exponential equations of the form ( a^x = b ) where (a) and (b) are positive numbers
• Rearrange exponential equations to isolate the variable using index laws
• Verify solutions in the original equations

Success Criteria

• Correctly solves at least 3 different exponential equations
• Explains the method used to isolate the unknown power
• Demonstrates understanding of index laws in their working

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Resources Required

• Whiteboard and marker, or paper and pen
• Calculator (scientific, with index and root functions)
• Set of prepared example questions and practice problems
• Index laws reference sheet (provided for support)

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Lesson Structure

1. Starter – Recall the Laws of Indices (5 minutes)

• Quick recap: student states the key index laws (product rule, quotient rule, power of a power) verbally or writes them down.
• Teacher asks the student to simplify examples such as ( 2^3 \times 2^4 ), ( \frac{5^7}{5^3} ), and ( (3^2)^4 ).
• Confirm understanding before moving on.

2. Introduction to Exponential Equations (10 minutes)

• Teacher introduces the form ( a^x = b ). Emphasise (a > 0), (a \neq 1), and (b > 0).
• Show how to solve if both sides can be expressed with the same base (e.g. ( 2^x = 8 \Rightarrow 2^x = 2^3 \Rightarrow x=3 )).
• Emphasise checking the reasonableness of answers.

3. Guided Practice: Solving Simple Equations (15 minutes)

• Work through 3 worked examples together with the student:
  1. ( 3^x = 27 )
  2. ( 5^{2x} = 125 )
  3. ( 4^{x+1} = 32 )
• Discuss each step, including rewriting bases where possible (e.g. (27 = 3^3), (125 = 5^3), (32 = 2^5)).
• Emphasise rewriting expressions to the same base before equating powers.

4. Independent Practice (20 minutes)

• Provide 5 problems to solve independently, varying difficulty slightly:
  1. ( 2^x = 16 )
  2. ( 9^{x-1} = 81 )
  3. ( 7^{2x+1} = 343 )
  4. ( 8^{x-2} = 1 )
  5. ( 10^x = 1000 )
• Student solves these, verbally explaining their thought process as they work through the problems.
• Teacher provides subtle scaffolding or prompts if needed (e.g. "Can you write 81 as a power of 9?").

5. Extension/Challenge (optional based on student ability):

• Introduce questions where the bases cannot be rewritten as integer powers directly (variant for stretch): ( 2^x = 20 ) (Introduce the idea of logarithms informally as a future topic; no solving needed).
• Have a discussion about how exponential equations become more complex.

6. Plenary and Assessment (10 minutes)

• Student to solve 3 exponential equations unassisted (mini-quiz):
  1. ( 6^x = 36 )
  2. ( 3^{x+2} = 81 )
  3. ( 5^{3x} = 125 )
• Student explains each answer in detail, demonstrating the process and justifying solutions.
• Teacher uses this to assess success criteria.
• Discuss any misconceptions or errors.

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Differentiation

• For deeper understanding, student can express more complex expressions where powers include binomials (e.g. ( (2^x)^3 )).
• For support, provide a detailed index laws sheet and step-by-step scaffolding questions.

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Homework (optional)

• Solve 5 more exponential equations (provided worksheet).
• Reflect in a short paragraph: “Why is it important to express numbers with the same base when solving exponential equations?”

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Assessment and Feedback

• Formative checks throughout (questioning, explanation).
• Final plenary problems serve as summative check of success criteria.
• Immediate oral feedback on working and solutions.
• Written comments on homework (if assigned).

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Teacher's Notes

• Because this is a 1:1 class, personalise examples using contexts the student enjoys where possible (e.g., growth and decay, computer science).
• Use mini-whiteboards or an online tool to dynamically solve and manipulate exponents.
• This lesson builds foundational skills needed for logarithms, which will appear later in GCSE content.

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National Curriculum Alignment Summary

NC Strand	Activity Link
Use and interpret algebraic notation	Writing and manipulating exponential equations
Laws of indices	Applying index laws to solve and simplify equations
Solve equations involving powers	Rearranging exponential equations to find unknown exponent

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This lesson plan provides a focused, scaffolded approach to solving exponential equations specifically designed for a Year 10 individual learner, closely aligned to the National Curriculum for England and ensuring clear, measurable outcomes.