At a glance. One lesson of about 60 minutes. It assumes lesson 2 and the index laws. Students derive the product, quotient and power laws from the definition, combine and split logarithms, and use them in equations where a solution must be checked against the domain.
Calculator: no calculator. No calculator: the laws are exact and the numbers are chosen so that each combined number is a power of the base. A calculator would hide the structure.
What students will learn
Know
- logb (mn) = logb m + logb n.
- logb (m ÷ n) = logb m − logb n.
- logb (mk) = k logb m.
Understand
- Why each law is an index law read through the definition.
- Why a sum inside a logarithm does not split.
- Why a solution of an equation in logarithms must keep every argument positive.
Do
- Combine two or more logarithms into one.
- Evaluate a combination of logarithms exactly.
- Solve an equation in logarithms and reject a root that makes an argument negative.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the three laws (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say what is always true of the joined bar.
- Criterion C. Communicating: writing the derivation of one law from the definition.
- Criterion D. Applying mathematics in real-life contexts: using a table of logarithms to avoid a long product (hook).
The one thing to do with the figure
Ask for the prediction first. Then set the base to 2, the first power to 0, and move the second power, and ask what the product is when m = 1 and why its logarithm equals log n.
Ask: “Why is the joined bar exactly as long as the second bar when the first power is 0?” The number m is 1, multiplying by 1 changes nothing, and log 1 is 0.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the library table; collect predictions about the product. |
| 12 min | Figure, key idea and the derivation card; write the three laws side by side with the index laws. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, ask what log2 8 is and whether the law adds or multiplies it. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 4; 28; 56. |
|---|---|
| Hook | The product 64 × 32 is 26 × 25 = 211, so the logarithm is 6 + 5 = 11. |
| Worked examples | 2; 3; loga 45. |
| Faded example | 4. |
| 1. Splitting a logarithm of a product | C. A and B turn the 8 into a power; D multiplies the logarithms. |
| 2. Product law | 5. log2 32. |
| 3. Quotient law | 7. log3 2187. |
| 4. Power law with a quotient | 10. log2 1024. |
| 5. Using given logarithms | 1.6. 3 × 0.3 + 0.7. |
| 6. Equation with two logarithms | 9. Reject −3. |
| 7. Equation with a difference of logarithms | 0.25. x + 1 = 5x. |
| 8. A sum inside a logarithm | A. B splits the sum; C multiplies the logarithms; D stops at the number. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Turned the factor into a power, or multiplied the logarithms. | Substitute x = 2: log2 16 = 4. Which option gives 4? |
| 11.4 or 32 in question 2 | Added the numbers inside, or gave the product without the logarithm. | Write 2 to the power of each logarithm and multiply: what power is the result? |
| 4372 or 2187 in question 3 | Subtracted the numbers, or stopped at the quotient. | Ask: is the quotient the answer, or the number the logarithm is of? |
| 1024 or 93.75 in question 4 | Stopped at the number, or multiplied the number by 2. | Ask: what does the 2 in front of the logarithm become when it moves inside? |
| 1 or 3.1 in question 5 | Used 2 and 5 only, or multiplied by 8. | Factorize 40 into primes and count each one. |
| −3 or 33 in question 6 | Kept a root that makes a logarithm undefined, or set one factor equal to 27. | Substitute each answer into the original equation. |
| 4 or 0.2 in question 7 | Inverted the fraction, or divided 1 by 5. | Substitute each answer into (x + 1) ÷ x and see if it gives 5. |
| B, C or D in question 8 | Split a sum as if it were a product, multiplied logarithms, or gave the number. | Work out 16 + 16 first, then ask what power of 2 it is. |
Questions worth asking
- Why does adding logarithms multiply the numbers? Listen for: adding powers of the same base is the same as multiplying the numbers.
- Is log m + log n the same as log (m + n)? Listen for: no, a counter-example with numbers.
- What does the joined bar say about the product law? Listen for: the bar for the product is the two logarithm bars end to end, so the logarithms add.
- Why must x − 6 be positive in question 6? Listen for: only positive numbers have a logarithm.
Differentiation
A way in for anyone stuck
- Give a table of m, n, log m, log n, mn and log mn for base 2 to fill in, so the pattern can be seen.
- Provide the three index laws on a card beside the three logarithm laws for matching.
- Give the frame: “Adding logarithms means the numbers are …. Subtracting means the numbers are ….”
Stretch
- Show that logb (1 ÷ n) = −logb n using the quotient law.
- Solve log2 (x + 3) + log2 (x − 3) = 4 and explain which root is rejected.
Students learning in English as an additional language
“Product” means the answer to a multiplication and “quotient” means the answer to a division. “Law” here means a rule that is always true. “Argument” means the number inside a logarithm. “Combine” means to join into one.
| Word | Plain meaning | Example |
|---|---|---|
| product law | the rule that gives the logarithm of a product from the logarithms of its factors | finding log of 6 × 7 from log 6 and log 7 |
| quotient law | the rule that gives the logarithm of a quotient from the logarithms of its parts | finding log of 20 ÷ 4 from log 20 and log 4 |
| power law | the rule that gives the logarithm of a power from the logarithm of its base | finding log of 5 squared from log 5 |
| argument | the number inside a logarithm, which must be positive | the 9 in log base 3 of 9 |
| combine | to write several logarithms as one | log 2 + log 7 as a single logarithm |
Sentence starters for explaining aloud or in writing:
- Adding the logarithms means I multiply …
- I wrote the number as a power of …, so the logarithm is …
- I rejected … because the argument would be …
Two things not to say
“Log of a sum is the sum of the logs.” Say instead: the sum law is for products; a sum inside has no simple rule.
“The log law lets you cancel the logs.” Say instead: combine the logarithms into one, then use the definition.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindWrite log7 5 + log7 3 as one logarithm.
log7 15. - FindEvaluate log10 50 + log10 2.
2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA table gives log2 10 = 3.32. Find log2 1000.
3 × 3.32 = 9.96. - Earlier work (Year 10)Write 34 × 32 as a single power.
36.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve log2 x = 3 log2 2 + log2 5.
x = 8 × 5 = 40. - Working backwardslogb 12 = p and logb 4 = q. Find logb 3.
p − q.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.