The unit starts from growth by a factor. A colony that doubles every hour is out of reach of adding, and the question "how many hours?" is the question "what power of 2?". The answer is a whole number only when the target is a power of the base, and a logarithm is the name for the power in every case, so the logarithm is introduced as the inverse of an exponential and evaluated by hand for powers of the base.
The three laws of logarithms are derived by writing each number as a power of the base and using the index laws, so they are not rules to be learned separately. They make exponential equations solvable exactly, by matching bases or by naming the power, and the change of base law gives a decimal value with a calculator. The same logarithms explain why acidity, earthquakes and loudness are reported on scales where one step is a factor of ten.
The last two lessons turn to inequalities. A quadratic changes sign only at its roots, so a sign diagram or a sketch gives the solution set. Rational and cubic inequalities follow the same plan once everything is on one side as a single fraction or product, with the values that make a denominator zero excluded.
The lessons, in order
- Lesson 1 · Exponential growth and "what power?" Exponential growth and decay as repeated multiplication, telling a constant ratio from a constant difference, and counting powers to find the step at which a target is reached. Choose a base and a target, and see which power first reaches it and whether it lands exactly. No calculator
- Lesson 2 · The logarithm as the inverse of a power The definition of a logarithm as the inverse of a power, the values log of 1 and log of the base, negative and fractional answers, and why zero and negative numbers have no logarithm. Choose a base and a power, and see the power fact and the logarithm fact as one arrow read both ways. No calculator
- Lesson 3 · The laws of logarithms The product, quotient and power laws of logarithms derived from the index laws, the values log of 1 and log of the base, and using the laws to combine, simplify and solve. Choose a base and two powers, and see that the bar for the product is the two logarithm bars joined end to end. No calculator
- Lesson 4 · Solving exponential equations Solving a power equation by matching bases, by writing the unknown as a logarithm, and by treating a power in a quadratic as a new unknown, with a bracket between whole numbers. Choose a base and a target, and see where the curve of powers crosses the target line. No calculator
- Lesson 5 · Change of base and calculator logarithms The change of base law derived from the definition, logarithms to any base with a calculator, solving exponential equations numerically, and counting whole periods to reach a target. Choose a base and a number, and see how many copies of the logarithm of the base fit into the logarithm of the number. Scientific calculator
- Lesson 6 · Logarithmic scales: sound, earthquakes and acidity Reading a logarithmic scale as an exponent, the ratio of two amounts from a difference in readings, decibels from a ratio of intensities, and combining two equal sounds. Choose two readings, and see how far apart the amounts are on a straight line and on a logarithmic line. Scientific calculator
- Lesson 7 · Quadratic inequalities and sign diagrams Solving quadratic inequalities with a sign diagram or a sketch, writing the solution set, handling an opening downwards and counting integer solutions, and using the discriminant to bound a coefficient. Choose two roots and the direction the parabola opens, and see which stretches of the axis are positive and which are negative. No calculator
- Lesson 8 · Rational and other non-linear inequalities Solving rational inequalities by combining into one fraction and reading a sign diagram, excluded values, cubic inequalities, and checking with the graphs of the two sides. Choose where the numerator is zero and where the denominator is zero, and see where the fraction is positive. GDC optional
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: logarithms, laws and equations | A | After lesson 4 | 25 minutes |
| B1: Investigation: powers and their length | B | Mid-unit, after lesson 5 | 45 minutes |
| C1: Explain it to someone who missed it | C | After lesson 8 | 35 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 8 | 45 minutes |
| D1: Sound at the night market | D and C | End of unit, after lesson 8 | 45 minutes |
| D2: The longtail boat cooperative | D and C | End of unit, after lesson 8 | 45 minutes |
D1 and D2 are alternatives. Both assess criteria D and C, in two different real-life contexts: one is set, not both.
Task A1: Quick check: logarithms, laws and equations (criterion A)
When: After lesson 4. Time: 25 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Answer every question you can and show your working: a correct method earns credit even if the final answer slips. Questions 11 and 12 are meant to make you think, so try something rather than leaving them blank.
Part 1. Evaluate
- Work out log2 32. (1 mark)
- Work out log3 (1 ÷ 9). (1 mark)
- Work out log10 0.001. (1 mark)
- Work out log6 4 + log6 9. (2 marks)
- Work out log2 24 − log2 3. (2 marks)
Part 2. One logarithm
- Write 2 log5 x + log5 y as a single logarithm. (2 marks)
Part 3. Equations
- Solve 3x = 81. (1 mark)
- Solve 52x − 1 = 125. (2 marks)
- Solve 4x = 8. (2 marks)
- Solve log2 (x + 1) = 4. (2 marks)
- Solve 9x = 3x + 4. (2 marks)
Part 4. Think it through
- Lena says: “log (a + b) is always the same as log a + log b.” Choose a base and values for a and b that show Lena is wrong, and show why they do. (2 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You evaluate a few simple logarithms and solve an equation where the answer is a whole number, with some working. |
| 3–4 | You evaluate logarithms of powers and use one law correctly, and solve equations by writing both sides as a power of the same base. |
| 5–6 | You use all three laws accurately, solve each equation with clear steps, and make a start on the two unfamiliar questions. |
| 7–8 | You choose the right law each time, solve every equation including the unfamiliar ones, and give a counterexample that you show really fails. |
Task B1: Investigation: powers and their length (criterion B)
When: Mid-unit, after lesson 5. Time: 45 minutes. Calculator: scientific calculator. Criterion: B, investigating patterns.
The number 2100 means 100 twos multiplied together. Written out in full it is much too long for any calculator screen. You can still say how many digits it has, without ever seeing all of them. This task is about how.
Part 1. Collect results
- Work out the powers of 2 from 21 to 220. Record each power and how many digits it has, in a table you design yourself.
- Do the same for the powers of 3, from 31 to 312.
Part 2. Describe what you find
- Look at how the number of digits changes as the exponent goes up, for 2 and for 3. Describe what you notice, in words, as a rule another student could use.
- Use your rule to predict how many digits 240 and 330 have. Then check both with a calculator.
Part 3. Test your rule
- Choose a base of your own from 4 to 9. Collect enough of its powers to test whether your rule, or a rule like it, works. Say what stays the same and what changes.
Part 4. Explain
- How many digits does 2100 have? Show how your method gives the answer.
- Explain why your method works, and whether it could ever be a little out for a particular power.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you work out some powers, count their digits and say something you notice about how the digit count changes. |
| 3–4 | With help, you complete both tables and suggest a rule for how often the digit count goes up, for 2 or for 3. |
| 5–6 | You organize your results clearly, describe a rule that fits both bases, and use it to predict and check 240 and 330. |
| 7–8 | You organize your results, describe a rule that fits all your findings, make correct predictions for large powers, test it on a base you chose, and say why it works and where it could slip. |
Task C1: Explain it to someone who missed it (criterion C)
When: After lesson 8. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Mara was away for lessons 3 to 8. She copied this working from a classmate, and it has mistakes in it.
The classmate’s working
(a) log2 8 + log2 4 = log2 (8 + 4) = log2 12.
(b) Solve x2 − 2x − 15 > 0. (x − 5)(x + 3) > 0, so x > 5 or x > −3, so x > −3.
Write a one-page explanation for Mara. She should be able to follow it on her own, without asking anyone.
Part 1. Put it right
- Find the mistake in each part. For each one, say what was done, why it is wrong, and write the correct working.
Part 2. Teach the idea
- Explain to Mara how to combine two logarithms of the same base that are added. Use the two logarithms in (a) and a check that she can do by evaluating each logarithm separately.
- Explain how a sign diagram gives the solution of a quadratic inequality. Draw one for part (b), mark the roots, and say how to test which regions are positive.
Part 3. Check the answer
- Show Mara one way to check her final answer to (b) by substituting a value, and say which value you chose and why.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | You write down the corrected lines with some words; the page is hard to follow without you there to explain. |
| 3–4 | You show correct steps for each part with the main symbols right, and a diagram or check that supports them. |
| 5–6 | You lay the page out line by line, use notation consistently, and give a reason for each step so Mara could follow it alone. |
| 7–8 | You use correct notation throughout, link the diagram to the written steps, and justify each step and the check, so the page stands on its own as a short worked guide. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 8. Time: 45 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Show your working. A correct method earns credit even if the final answer slips. Give answers to 3 significant figures unless a question says otherwise. Questions 10 to 12 are different from the ones you have practised, so try something rather than leaving them blank.
Part 1. Laws of logarithms
- Work out log3 (1 ÷ 27). (1 mark)
- Write log 12 + log 5 − log 3 as a single logarithm. (2 marks)
- Write loga (x3 ÷ y) in terms of loga x and loga y. (2 marks)
Part 2. Exponential equations
- Solve 8x − 1 = 4, giving x as an exact fraction. (3 marks)
- Solve 3x = 20. (2 marks)
- Solve 5 × 2x = 70. (3 marks)
- Use the change of base law to evaluate log4 100. (2 marks)
Part 3. Inequalities
- Solve x2 − 5x − 14 ≤ 0. (3 marks)
- Solve (x − 3) ÷ (x + 2) > 0. (3 marks)
Part 4. Scales and applications
- A drink has pH 4.6 and a second drink has pH 6.1. How many times more acidic is the first? (3 marks)
- Show that log9 x = 0.5 log3 x. (3 marks)
- A population grows by 6% each year. After how many whole years does it first become more than double its starting size? (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You evaluate a simple logarithm, use one law, and solve an equation or inequality where the working is mostly set out for you. |
| 3–4 | You use the laws on familiar forms, solve both kinds of exponential equation, and solve one inequality with a sign diagram. |
| 5–6 | You work accurately across all four parts, give exact and rounded answers as asked, and make real progress on the unfamiliar questions. |
| 7–8 | You choose methods without prompting, solve the rational inequality with its excluded value, and complete the proof and the growth question with correct justification. |
Task D1: Sound at the night market (criteria D and C)
When: End of unit, after lesson 8. Time: 45 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A night market in Chiang Mai has 40 food stalls along one street. The market committee has a rule that the sound level at any stall must stay at or below 85 decibels. The sound level in decibels (dB) is L = 10 log10 (I ÷ I0), where I is the sound intensity in watts per square metre and I0 = 10−12 is the quietest sound a person can hear.
A street musician plays through an amplifier. At the nearest stall the intensity is 3.2 × 10−5 watts per square metre. Several musicians have asked to play in the same area, and the committee wants to know how many identical amplifiers can play together before the rule is broken.
Part 1. One amplifier
- Work out the sound level of one amplifier at the nearest stall, to 1 decimal place.
- A different amplifier is measured at exactly 85 dB. Work out its intensity in watts per square metre, as a number times a power of ten.
Part 2. Several amplifiers
- Two identical amplifiers play together at the same distance. Find the total sound level, and the increase compared with one amplifier.
- Find the largest number of identical amplifiers that can play together at the nearest stall without breaking the 85 dB rule.
Part 3. Advise the committee
- A stall holder says: “85 decibels is only a little louder than 75, so one extra amplifier cannot matter.” Use your results to say whether this is right, in a way a non-mathematician on the committee would follow.
- Write a short recommendation to the committee. State the model you used, one assumption in it that might not hold in a real street, and what you would measure to check it.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You use the decibel formula to find a level with some help, and say what your answer means for the market. |
| 3–4 | You find the level and the intensity correctly, and use the formula for two amplifiers, linking the results to the 85 dB rule. |
| 5–6 | You solve the whole problem with a clear model, find the largest number of amplifiers, and say what an assumption means for the result. |
| 7–8 | You solve every part, justify the choice of model, test an assumption against the result, and reach a recommendation the data really support. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | You write the answers with units, and some working a reader can see. |
| 3–4 | You show the working in order, with the formula and units written out, and a sentence that states each answer. |
| 5–6 | You explain each step so a committee member could follow it, and the recommendation refers to your numbers. |
| 7–8 | You tailor the explanation to the audience, state the model and its limits precisely, and keep notation and units consistent throughout. |
Task D2: The longtail boat cooperative (criteria D and C)
When: End of unit, after lesson 8. Time: 45 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A cooperative of boat operators on a lake wants to replace an old longtail boat engine. They have 120 000 crowns in an account that pays 3.5% interest each year, added to the account at the end of each year. A new engine costs 200 000 crowns. The old engine is worth 90 000 crowns now, and loses 12% of its value each year.
The cooperative also sets the ticket price p crowns for a lake trip. The daily profit in crowns, from a model fitted to last year’s trips, is P = −2p2 + 300p − 10 000.
Part 1. Saving for the engine
- After how many whole years will the account first hold at least 200 000 crowns? Show the equation you solve.
Part 2. The old engine
- After how many whole years will the old engine first be worth less than 30 000 crowns?
Part 3. Ticket price
- Find the range of ticket prices for which the cooperative makes a profit.
- The cooperative needs a daily profit of at least 800 crowns. Find the range of prices that achieve this, and the price that gives the largest profit.
Part 4. The plan
- Write a short plan for the cooperative. Say when they can afford the engine, what price to set, and one thing about the models that might make your advice wrong.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You set up an equation for the savings or the engine value and solve it by trial or with help, then say what the answer means. |
| 3–4 | You solve the savings equation with logarithms and find the range of prices that make a profit, with the steps shown. |
| 5–6 | You solve all four parts with a clear method, link each result to the cooperative’s decision, and name a limit of a model. |
| 7–8 | You solve every part with justified methods, check the answers against the original conditions, and give a plan that weighs the two models and their limits. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | You write your answers with units, and some working a reader can see. |
| 3–4 | You write each equation and inequality clearly, in order, with a sentence stating each answer in context. |
| 5–6 | You explain each step to someone on the cooperative committee, use a sign diagram or sketch for the price range, and use consistent notation. |
| 7–8 | You choose a presentation that suits the audience, keep the mathematics precise and the language plain, and make the link between the plan and the models explicit. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.