Math lesson plan · 11th grade (Year 12 in England) · 30 minutes
Expand a logarithm using the log laws
Algebra
CCSS HSF.LE.A.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Expand a logarithm using the log laws.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Combine logarithms into a single logarithm
Leads to: the next skill in this area.
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Combine logarithms into a single logarithm). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- log 6 + log 2 − log 3 = log N. Find N. → 4Prompt: Adding logs multiplies the numbers; subtracting divides.
- 3 log 3 + log 5 = log N. Find N. → 135Prompt: k log a = log aᵏ (power law).
- Write 3 log 3 − log 9 + log 4 as a single logarithm log N. Find N. → 12Prompt: Use the power law first.
Hook · 3 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Expand a logarithm using the log laws. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: a = log x, b = log y and c = log z. Write log(x⁴y/z²) in terms of a, b and c.
- Step 1: 4a + b − 2c
- Answer: 4a + b − 2c
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- a = log x, b = log y and c = log z. Write log(x⁴y/z) in terms of a, b and c. (4a × b ÷ c / 4a + b − c / 4ab − c / 4a + b + c) → 4a + b − cPrompt: Products become sums; quotients become differences.
- Omar says log(x²) = (log x)². Which is the correct rewrite of log(x²)? (log 2 + log x / (log x)² / 2 log x / log x ÷ 2) → 2 log xPrompt: Test with x = 10: log(10²) = 2.
You do: independent practice · 7 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Liam says log(x³) = (log x)³. Which is the correct rewrite of log(x³)? (3 log x / log x ÷ 3 / (log x)³ / log 3 + log x) → 3 log xPrompt: Test with x = 10: log(10³) = 3.
- a = log x, b = log y and c = log z. Write log(xy³/z³) in terms of a, b and c. (4a − 3c / a + 3b + 3c / a + 3b − 3c / a3b − 3c) → a + 3b − 3cPrompt: Products become sums; quotients become differences.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Combine logarithms into a single logarithm.
- Use concrete materials: balance scales or a balance drawing.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.
Free lesson plan from eurekami.com/lesson-plan-builder