Analysis only. It needs the logarithm laws from 1.7 and feeds the equation-solving at 2.10, where nearly every interesting equation has an exponential in it.
Drag the slider from x = 0 to x = 6 and watch the dashed gap open. At x = 1 and x = 2 it closes to nothing, which is the whole reason students write 2x for 2x: the two expressions really do agree, twice, and a class that has only ever seen small numbers has seen only the agreement.
Then say the two out loud, slowly: two to the x, and two x. Students who read them silently blur them, and naming the difference as a difference in words fixes more of this than any amount of algebra.
View 2 is the inverse relationship, with y = x dashed between the two curves and the pairs (3, 8) and (8, 3) marked. Point at one pair and ask for the other; the swap is the whole idea and it costs a sentence.
| Question | Answer |
|---|---|
| 1. 210 | 1024. |
| 2. 2 × 10 | 20. |
| 3. log28 | 3, because 2³ = 8. |
| 4. ln 2 | 0.693. |
| 5. Why every exponential passes through (0, 1) | B. a0 = 1 for every positive a. |
Questions 1 and 2 look insultingly easy and are the point of the page. Set them as a pair, in that order, and the 51.2 arrives from their own two answers rather than from you.
1 markThe right shape, with the asymptote drawn.
1 markThe special point, (0, 1) or (1, 0).
1 markThe domain or range, where asked.
On "sketch the inverse" the reflection in y = x is accepted working, and it is faster than rearranging. The mark needs the line drawn AND labelled; two curves on their own do not show a reflection.
| They wrote | What happened |
|---|---|
| 20 on question 1 | The confusion the page is for. They read 210 as 2 × 10. Put questions 1 and 2 side by side and the error becomes visible to them. |
| 100 | 10²: base and exponent swapped. Worth a word, because the same swap turns log28 into log82. |
| 512 or 2048 | 2⁹ or 2¹¹. An off-by-one in the counting, not in the method, and usually a student doubling on their fingers. |
| 12 | 2 + 10 on question 2. Rare, and it means the word "times" was not read. |
| 8 on question 3 | Gave the number inside. "Log of 8" has been read as "8". |
| 4 on question 3 | 8/2. A logarithm treated as a division. Ask which power of 2 gives 8, in those words. |
| 0.903 | log108, from the plain log key with the base ignored. The commonest calculator slip here. |
| 2.079 | ln 8. Same slip, other key. |
| 0.301 on question 4 | log102 rather than ln 2. The two keys sit next to each other and both look right. |
| 1.443 | 1/ln 2, which is log2e. They have inverted at the end. |
"Why e?" Because it is the base whose curve has gradient equal to its own height, which is what makes the calculus in Topic 5 clean. At this stage the honest answer is that it is coming, and that e is just a number, 2.718, the way π is just a number. Resisting a full answer here is better than a vague one.
"Is exponential growth just fast growth?" No, and the guide raises this as an Aim 8 question. Exponential means the step is proportional to the size, so it grows when a > 1 and shrinks when 0 < a < 1 with the size: 2x adds 1 from x = 0 to x = 1 and adds 512 from x = 9 to x = 10. A straight line can be far steeper than an exponential over a short window and be overtaken for ever afterwards. Worth five minutes with a news example, because the word is used loosely everywhere and students repeat the loose version.
"Can the base be negative?" Not for these functions. (−2)x has no value at x = ½, so there is no curve to draw. The guide restricts a > 0 for exactly that reason, and it is worth one line so the restriction is a fact rather than a rule.
"Why does ln of a negative not exist?" Because no power of e is negative, so nothing could be the answer. It matters in practice: solving an equation that produces ln(−3) means that root is rejected, and the rejection is a mark.
| Demonstrate | Graph 2x, log2x and y = x together in a SQUARE window. The square window is the whole demonstration; in the default rectangle the reflection looks like two unrelated curves and the idea is lost. Then e^(3*ln(2)) on a calculator page, returning 8, which is the base-change relationship in one entry. |
|---|---|
| Where they stick | The base. The plain log key is base 10 and the ln key is base e, so log28 needs the base entered: log(8,2) on the Nspire, logab(2,8) on the Casio, where the BASE comes first and the two orders give different answers without any complaint. Have them check against log28 = 3, which they can verify by hand. |
| The check | Undo it. If log28 is claimed to be 3, compute 2³ and see 8. One entry, and it catches the wrong base, the swapped arguments and the inverted answer. |
Analysis Paper 1 has no calculator, and these questions are chosen so the numbers are exact: log28 = 3, ln e = 1, ln 1 = 0. Set a batch with the machines away.
| Step | What |
|---|---|
| 1 | Questions 1 and 2 cold, in that order, on paper. Collect both answers. |
| 2 | Figure view 1. Drag from 0 to 6 and let the gap open. |
| 3 | Say both expressions out loud. Name the difference in words. |
| 4 | The shape of ax: through (0, 1), asymptote y = 0, never zero or negative. |
| 5 | Logarithms as the inverse, then view 2 with the square window. |
| 6 | The table of swapping features, read across rather than down. |
| 7 | ax = ex ln a, checked on the machine. |
| 8 | The Aim 8 discussion about "exponential" in the news, if there is time. |
Do not say "exponential means it grows really fast". It is the loose usage the guide asks you to question, and it leaves students unable to say what is actually different about it. Say that the step is proportional to the size, growing for a > 1 and shrinking for 0 < a < 1 with the size, and give them the 1 against 512.
Do not say "log is the opposite of power" and move on. It is true and unusable. The usable form is that logay is the power you raise a to in order to get y, because that sentence can be substituted straight into a problem. Say the long version until they can say it back.