Logarithmic Thinking

Lesson Overview

Subject Area: Mathematics
Key Stage: KS5 (Year 12 - A-Level)
Curriculum Reference: AQA A-Level Maths (Algebra and Functions) – Section 3.6 Exponentials and Logarithms
Learning Objective: Students will solve logarithmic equations using the laws of logarithms and apply these skills to contextual problems.
Duration: 80 minutes
Class Size: 30 students

---

Learning Outcomes

By the end of the lesson, students will be able to:

1. Recall and apply the laws of logarithms, including product, quotient, and power rules.
2. Solve logarithmic equations step by step with confidence and accuracy.
3. Analyse and solve real-life situations involving logarithmic relationships.

---

Resources

• Whiteboard and markers
• Scientific calculators for all students
• Mini-whiteboards for student engagement
• A set of context-based logarithmic problem cards (teacher-prepared)
• Digital visualiser or projector for class demonstration
• A short, engaging starter quiz (paper or projected slide)

---

Lesson Structure

Time Breakdown:

• Starter: 10 mins
• Direct Instruction & Worked Examples: 20 mins
• Guided Practice: 20 mins
• Application and Problem Solving: 20 mins
• Plenary: 10 mins

---

1. Starter (10 mins)

Activity: “Logarithmic Warm-Up Challenge”

• Students complete a short 5-question multiple-choice quiz where they recall and apply basics of logarithms. This includes evaluating logarithms (e.g., log₂8, log₁₀1000) and converting between exponential and logarithmic forms.
• Use mini-whiteboards for answers. The teacher introduces an element of gamification by using a fast-paced buzzer countdown when students hold up their mini-whiteboards.

Purpose:

• Diagnostic tool to assess prior knowledge and identify misconceptions.
• Engages students and creates curiosity.

---

2. Direct Instruction & Worked Examples (20 mins)

Teacher Explanation (10 mins):

1. Recap key laws of logarithms:
   • Product Rule: logₐ(xy) = logₐx + logₐy
   • Quotient Rule: logₐ(x/y) = logₐx − logₐy
   • Power Rule: logₐ(xⁿ) = n * logₐx
2. Discuss common pitfalls (e.g., assuming logₐ(x + y) = logₐx + logₐy).
3. Use relatable, real-world examples (e.g., the Richter scale for earthquakes or pH levels in chemistry).
4. Write and solve a simple equation step by step:
   Example: Solve log₂(x − 3) + log₂(x + 3) = 4
   Solution: Combine the logs using the product rule, solve quadratic.

Live Worked Examples (10 mins):

• The teacher works through 2 progressively difficult logarithmic equations, projected for clarity.
• Emphasis on annotation and breaking problems into logical steps.
• Example 1: Solve log₃(x) = 4.
• Example 2: Solve log₅(x + 1) − log₅(x − 1) = 1.

---

3. Guided Practice (20 mins)

Paired Activity – “Collaborative Error Spotting”

• Students work in pairs and are provided 4 solved logarithmic equations with intentional errors.
• Their task is to identify the errors, correct them, and explain their reasoning.
• Example problem with error: log₁₀(x²) = log₁₀(x) * log₁₀(x) (error: incorrect application of power rule).

Purpose:

• Consolidates learning through peer discussions.
• Encourages critical thinking and collaboration.

---

4. Application and Problem Solving (20 mins)

Activity: “Logarithms in Real Life”

Context Cards: Each group of students (in groups of 4) is given a scenario-based logarithmic question. Examples include:

1. Earthquake Magnitude: "The magnitude of an earthquake is measured by M = log₁₀(I/I₀), where I is the earthquake's intensity. If I = 10,000I₀, find M."
2. Population Growth: "In a population following exponential growth, log₂(P) = 5, where P is the population size. Solve for P."
3. pH Levels in Chemistry: "The pH of a solution is measured by pH = -log₁₀[H⁺]. If pH = 3, find [H⁺]."

Procedure:

• Each group tackles one card and explains their solution to the class.
• Encourage groups to critique and compare approaches.

Purpose:

• Connects mathematical theory to real-world concepts.
• Develops problem-solving, communication, and reasoning skills.

---

5. Plenary (10 mins)

Quickfire Quiz – “Laws and Logic”

• Students answer 5 rapid-fire questions on mini-whiteboards to recap:
  1. What is log₄16?
  2. Simplify: log₃9 + log₃27.
  3. True or False: log₆(2) + log₆(3) = log₆(5).
  4. If log₅x = 2, find x.
  5. Solve: log₂(x²) = 6.

Closing Reflection:

• Hand out exit tickets where students write down 1 thing they’ve learned and 1 question they still have.

---

Differentiation

• For High Achievers: Include more complex problems, such as equations involving change of base formula:
  log₆x = log₅x / log₅6.
• For Struggling Students: Provide scaffolded prompts (e.g., rearranged equations or partial solutions). Offer support in smaller groups during guided practice.

---

Homework Task

• Complete a worksheet with logarithmic equations and contextual problems to reinforce today’s lesson.
• Challenge: Research and write about one real-life application of logarithms not covered in class.

---

Assessment for Learning

1. Observe and address misconceptions during the starter activity and plenary quiz.
2. Monitor group discussions during collaborative tasks to assess understanding.
3. Analyse exit tickets to inform planning for the next lesson.

---

Extension

STEM Link Challenge:

• Research how logarithms are used in sound engineering or in measuring the brightness of stars (magnitude scale). Write a one-page report summarising findings using mathematics to support examples.

---

This highly interactive and applied lesson ensures students develop mastery of logarithmic equations while connecting abstract concepts to practical, real-world contexts – a clear way to engage Year 12 learners.