At a glance. One lesson of about 60 minutes. It assumes lessons 3 and 4. Students derive the law from the definition, apply it with the base 10 key, estimate before using the calculator, and solve unknowns that appear in two different powers.
Calculator: scientific calculator. A scientific calculator is used because the arithmetic is the point here: the law reduces any logarithm to two key presses, and an estimate with whole powers is the check that the keys were right.
What students will learn
Know
- logb N = log N ÷ log b, with any base for both logarithms.
- logb a = 1 ÷ loga b.
- bx = N gives x = log N ÷ log b.
Understand
- Why the law follows from the power law applied to bx = N.
- Why the number is on top: it is the longer logarithm in the figure.
- Why growth problems ask for the next whole period, not the nearest.
Do
- Evaluate a logarithm to any base to a given precision.
- Solve bx = N and equations with unknowns in two powers.
- Round a time up to the whole period in which a target is passed.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the law and its use (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say how the copies change as the base grows.
- Criterion C. Communicating: writing the substitution log N ÷ log b and the rounding step.
- Criterion D. Applying mathematics in real-life contexts: savings, population and a fund (hook and questions 4 and 5).
The one thing to do with the figure
Ask for the prediction first. Then set the base to 2 and the number to 64, and ask how many copies of log 2 fit into log 64 and why the answer is a whole number.
Ask: “What happens to the number of copies as the base increases for the same number?” The base copy gets longer, so fewer copies fit.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the savings account; collect predictions. |
| 12 min | Figure, key idea and the warning card; derive the law from bx = N. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, ask which number the top logarithm belongs to; if D, point to the base c in the key card. |
| 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 | log2 60; 7; 260. |
|---|---|
| Hook | 1.05t = 2 gives t = 0.30103 ÷ 0.021189 = 14.2 years, so money doubles in a little over 14 years. |
| Worked examples | 2.72; about 11.9 years; 2.63. |
| Faded example | 3.61. |
| 1. Reading the change of base law backwards | A. B is the reciprocal; C divides inside the logarithm; D thinks the law needs base 10. |
| 2. Evaluating a logarithm to any base | 2.93. log 300 ÷ log 7. |
| 3. Solving an exponential equation numerically | 4.07. log 700 ÷ log 5. |
| 4. Growth to a target in whole years | 18. t = 17.28. |
| 5. Time to triple | 17. t = 16.24. |
| 6. Unknown in both powers | 2.71. x = log 3 ÷ (log 3 − log 2). |
| 7. Unknown in both powers, with a negative answer | −1.35. x = −log 3 ÷ (2 log 3 − log 4). |
| 8. Estimating before using the calculator | C. A is the reciprocal; B is log 400 only; D divides 400 by 3. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Put the base on top, divided inside the logarithm, or thought the law needs base 10. | Ask: in logc N ÷ logc b, which is the number and which is the base? |
| 0.34 or 1.63 in question 2 | Inverted the fraction, or divided inside the logarithm. | Use the estimate: 73 = 343 is just above 300. |
| 140 or 0.25 in question 3 | Divided 700 by 5, or inverted the fraction. | Check by substituting: does 50.25 give 700? |
| 17 or 23 in question 4 | Rounded down, or used a linear model. | Calculate the population after 17 years and compare with 20 000. |
| 16 or 29 in question 5 | Rounded down, or used simple growth. | Calculate 1.0716 and compare with 3. |
| 5.68 or 0.37 in question 6 | Forgot log 3 on the right, or inverted. | Substitute x = 2.71 into both sides to compare. |
| 1.35 or −2.84 in question 7 | Dropped the minus sign, or forgot log 3 on the top. | Ask: is x positive or negative if 32x + 1 must equal 4x? |
| A, B or D in question 8 | Gave the reciprocal, gave log 400 only, or divided 400 by 3. | Use the whole-power bracket: which two powers of 3 surround 400? |
Questions worth asking
- Why does the number go on top? Listen for: the question is how many copies of log b fit into log N.
- Does it matter if we use log or ln? Listen for: no, if the same key is used on the top and the bottom.
- Why round up when finding the number of years to pass a target? Listen for: the target is only passed at the end of a whole year.
- What is logb b, and does the law give it? Listen for: 1, since log b ÷ log b = 1.
Differentiation
A way in for anyone stuck
- Give a fill-in line: logb N = log … ÷ log … = … ÷ … = ….
- Provide a table of base 10 logarithms for 2, 3, 5, 7 to use with the law.
- Give the frame: “The number is … and the base is …, so I type ….”
Stretch
- Show that log4 N = (log2 N) ÷ 2 for every N.
- Find the smallest whole number n for which 0.9n is less than 0.05.
Students learning in English as an additional language
“Change of base” means writing a logarithm with a different base. “Round up” means to go to the next whole number even if the decimal part is small. “Compound” means that the growth is applied to the new amount each time. “Exceed” means to be more than.
| Word | Plain meaning | Example |
|---|---|---|
| change of base | writing a logarithm as a quotient of two logarithms with a new base | log 9 ÷ log 4 |
| natural logarithm | the logarithm to the base called e, written ln | the ln key |
| compound growth | growth applied to the amount at the end of each period | 5% a year on the new total |
| round up | to go to the next whole number even if the first decimal is small | 11.9 years becomes 12 years |
| exceed | to be greater than | more than 20 000 people |
Sentence starters for explaining aloud or in writing:
- I put the number on top because …
- My estimate was … and my calculator gave …
- I rounded up because …
Two things not to say
“Just divide the logs.” Say instead: the number is the longer logarithm and goes on top.
“Round to the nearest.” Say instead: round up when the target must be passed.
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.
- FindFind log9 700 to two decimal places.
2.98. - FindSolve 2x = 50 to two decimal places.
5.64.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone loses 15% of its value each year. After how many whole years is it worth less than half of its start?
0.85t < 0.5 gives t = 4.27, so 5 years. - Earlier work (Year 10)Increase 80 by 5% twice.
88.2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWithout a calculator, put log2 10 × log10 2 in its simplest form.
1. - Working backwardsGiven that loga 10 = 2.5, find log10 a.
1 ÷ 2.5 = 0.4.
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.
A scientific calculator is needed for questions 2 to 7.