Mathematics from zero
Growth: multiple-choice review
Crux Multiple-choice synthesis across the growth unit — telling adding from multiplying, why exponential overtakes linear, and reading a logarithm as the exponent that reverses it.
Your altitude — climbing toward senior
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You are at middle altitude — in the skySix questions that span the whole unit. Each one is a small decision — is this adding or multiplying, will it creep or explode, what does this logarithm actually ask — not a definition to recite.
Goal
Confirm you can tell linear growth from exponential growth, explain why multiplying always overtakes adding in the long run, and read a logarithm as the exponent that reverses an exponential.
Quiz
Completed
Jar A starts at 0 and adds 8 each day. Jar B starts at 1 and doubles each day. On day 3, A holds 24 and B holds 8. Which jar is bigger far out at day 10, and why?
Heads-up Being ahead early says nothing about the long run. Exponential growth often starts slower, then its step outgrows any fixed amount and it pulls away.
Heads-up Both grow, but by different rules. A adds a fixed 8; B multiplies, so its growth accelerates. Equal daily change is not what is happening.
Heads-up It is a bigger step only at the very start. Once B's total passes 8, doubling adds more than 8 per day, and the gap only widens.
Quiz
Completed
Which of these grows exponentially rather than linearly?
Heads-up Adding the same fixed amount each step is linear growth, not exponential. Exponential growth multiplies by a fixed amount.
Heads-up A steady fixed amount per unit of time is linear. The level rises by the same 2 litres each minute — no acceleration.
Heads-up Constant speed adds the same distance each minute, which is linear. Nothing is being multiplied.
Quiz
Completed
A logarithm log₂(32) = 5 means what?
Heads-up A logarithm is not division. 32 ÷ 2 is 16, not 5. log₂(32) asks for the exponent that reaches 32, which is 5 because 2⁵ = 32.
Heads-up A logarithm is not multiplication of the two numbers — and 2 × 5 is 10, not 32. It is the exponent connecting the base to the target.
Heads-up A logarithm has nothing to do with subtraction. It answers '2 to what power is 32?' — the answer is the exponent, 5.
Quiz
Completed
Starting from 1 and doubling, the totals are 1, 2, 4, 8, 16, 32, 64. How many doublings does it take to first reach 64, and how does that connect to a logarithm?
Heads-up Doubling is not adding one at a time. Each doubling multiplies by 2, so reaching 64 takes only 6 steps, not 64.
Heads-up Going from 32 to 64 is a single doubling, the sixth one. The whole climb from 1 to 64 is 6 doublings, so log₂(64) = 6.
Heads-up The starting 1 is 2⁰ — zero doublings. Reaching 64 = 2⁶ takes 6 doublings, giving log₂(64) = 6.
Quiz
Completed
Why does the log₂ curve climb so slowly — as the input runs from 1 to 64, the output only rises from 0 to 6?
Heads-up The logarithm keeps rising, never shrinking — it just rises very slowly. Each doubling of the input adds 1, so by 64 it has only reached 6.
Heads-up Logarithms handle huge numbers fine — that slowness is exactly why they are useful for measuring explosive growth on a manageable scale.
Heads-up It does not stop. Double the input again to 128 and the log reaches 7. It climbs forever, just one step per doubling.
Quiz
Completed
A rumour spreads by each person telling one new person each hour, starting from 1 person. Which description is correct?
Heads-up Each informed person tells someone new, so the number of tellers grows every hour. That is multiplying, not adding a fixed amount — exponential, not linear.
Heads-up Spreading person-to-person multiplies the count, so it is exponential growth. The logarithm is only the reverse measuring tool, not a later phase of the spread.
Heads-up It is a textbook case of exponential growth: a quantity that multiplies each step, exactly like doubling money or a population.
Recap
The through-line of the unit: linear growth adds a fixed amount each step, exponential growth multiplies by a fixed amount — so exponential always overtakes linear in the long run, even when it starts slower. A logarithm runs that exponential backwards: log₂(N) is the exponent on 2 that reaches N, which is roughly how many doublings it took. That is why the log climbs by just 1 per doubling and turns explosive growth into a small, calm number.