At a glance. One lesson of about 60 minutes. It assumes negative and fractional indices from Year 10. Students translate between power and logarithm statements, evaluate logarithms by writing the number as a power of the base, and say why zero and negative numbers have no logarithm.
Calculator: no calculator. No calculator: every number here is a power of the base, and the skill is recognising it. Calculator logarithms of other numbers come in lesson 5.
What students will learn
Know
- logb N = x means bx = N.
- logb 1 = 0 and logb b = 1.
- The base is positive and not 1, and N is positive.
Understand
- Why a logarithm is the inverse of a power: each undoes the other.
- Why numbers below 1 have negative logarithms for a base above 1.
- Why zero and negative numbers have no logarithm.
Do
- Write a power fact as a logarithm fact and back.
- Evaluate a logarithm by expressing the number as a power of the base.
- Solve a simple equation in a logarithm for the number or the base.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: converting between the two forms and evaluating (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say what stays the same for every base when the power is 0.
- Criterion C. Communicating: writing a logarithm statement and its power statement together.
- Criterion D. Applying mathematics in real-life contexts: recording the size of a number by its power of ten (hook).
The one thing to do with the figure
Ask for the prediction first. Then set the power to 0 and move the base from 2 to 10, and ask what the number is each time and what the logarithm is.
Ask: “What does the bottom arrow do that the top arrow does not?” It takes the number back to the power it came from.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the warehouse record of tens; collect predictions. |
| 12 min | Figure, key idea and warning card; write each power fact as a logarithm fact. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or C, ask which number sits after the log sign. |
| 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 | 6; 2−3; 7. |
|---|---|
| Hook | The record is 5 for one hundred thousand, and for 0.001 it is −3, because 0.001 = 10−3. |
| Worked examples | 4; −4; 0.3333; 3. |
| Faded example | −3. |
| 1. Reading a power as a logarithm | A. B uses the power as the base; C swaps the base and the number; D swaps the number and the logarithm. |
| 2. Evaluating a logarithm | 5. 25 = 32. |
| 3. Logarithm of a decimal | −2. 0.01 = 10−2. |
| 4. Finding the base | 4. x5 = 1024. |
| 5. Fractional logarithm | 0.5. 360.5 = 6. |
| 6. Solving a logarithm statement | 0.0625. 4−2 = 1 ÷ 16. |
| 7. Finding a base from a fractional logarithm | 49. 49 × 7 = 343. |
| 8. Which logarithm statements are true | B. A gives a negative number a logarithm; C swaps base and number (it is 0.5); D confuses log 0 with log 1. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Swapped the base, the number or the answer. | Ask: which number is multiplied by itself, and which counts the times? |
| 16 or 25 in question 2 | Divided 32 by 2, or swapped base and answer. | Write 2, 4, 8, 16, 32 and count the entries. |
| 2 or 100 in question 3 | Dropped the negative, or gave the reciprocal's denominator. | Ask: is 0.01 bigger or smaller than 1, and so is the power positive or negative? |
| 5120 or 204.8 in question 4 | Multiplied or divided by the logarithm. | Write the power form, x5 = 1024, and try 3 and then 4. |
| 6 or 2 in question 5 | Divided 36 by 6, or evaluated log6 36. | Ask: is the power more or less than 1 when the answer is smaller than the base? |
| −8 or 16 in question 6 | Multiplied 4 by −2, or dropped the minus sign. | Ask for 4−1 first and then 4−2. |
| 7 or 514.5 in question 7 | Took the cube root of 343, or multiplied by 1.5. | Check by substituting: does 71.5 give 343? |
| A, C or D in question 8 | Gave a negative number a logarithm, swapped base and number, or mixed up log 0 and log 1. | Write each statement as a power and test it. |
Questions worth asking
- Why is log of 1 equal to 0 for every base? Listen for: any number to the power 0 is 1.
- Is log2 8 bigger or smaller than log2 16, and why? Listen for: smaller, because a bigger number needs a bigger power.
- Why can the answer be negative when the number is not? Listen for: the number is below 1, which needs a negative power.
- What is logb of b, and why? Listen for: 1, because b to the power 1 is b.
Differentiation
A way in for anyone stuck
- Give a list of powers of 2, 3, 5 and 10 up to 1000 and a table with the headings base, power and number.
- Provide cards with a power fact on one side and the matching logarithm fact on the other.
- Give the frame: “The base is …. To make … I raise it to the power …, so the logarithm is ….”
Stretch
- Show that log4 8 = 1.5 by writing both as powers of 2.
- Explain why there is no base for which logb 5 = 0.
Students learning in English as an additional language
“Logarithm” is the name for the power. “Base” is the number that is raised to a power. “Inverse” means that it undoes another operation. “Evaluate” means to work out the value.
| Word | Plain meaning | Example |
|---|---|---|
| logarithm | the power a base is raised to in order to make a number | log base 6 of 36 |
| base | the number that is multiplied by itself in a power | the 6 in 63 |
| inverse | the operation that undoes another one | taking a square root undoes squaring |
| evaluate | to work out the value of an expression | log base 7 of 7 |
| reciprocal | one divided by a number | the reciprocal of 10 is 0.1 |
Sentence starters for explaining aloud or in writing:
- The base is … and the number is …, so the power is …
- This is negative because the number is …
- I wrote the number as a power of …
Two things not to say
“A logarithm is a button on the calculator.” Say instead: a logarithm is a power, and most of today's can be found by listing.
“The log of a negative number is negative.” Say instead: negative numbers have no logarithm to a positive base.
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.
- FindEvaluate log6 216.
3. - FindEvaluate log9 1.
0.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe record of a number in tens is log10 of it. A number has record 6. What is the number?
1 000 000. - Earlier work (Year 10)Write 1 ÷ 32 as a power of 2.
2−5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind b if logb 0.25 = −2.
b−2 = 1 ÷ 4, so b = 2. - Working backwardslog3 N = −1. Find N.
1 ÷ 3.
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.