Compound interest, half-life, log scales, and freeing an exponent with a log — exponentials in the world.
The exponential is not a curiosity for the chalkboard — it is how money compounds, how radioactivity fades, how a rumor or a virus spreads, and how we tame numbers that range across a dozen powers of ten. In this lesson we point the tools you have built at the real world. We model growth with A = P(1 + r)n and decay with A = A0(½)t/T, then use a logarithm to free an unknown exponent — to answer "when does it double?" We squeeze earthquakes, sound, and acidity onto a log scale, where equal steps mean equal multiples. And we feel, in the bone, just how violent exponential growth really is. Every number here is computed from GX.expf, GX.logb, and the plane engine, so each picture tells the truth.
Put P dollars in an account that pays interest rate r each period and leaves it to grow. After one period the balance is P(1 + r). The next period earns interest on that new total, so it becomes P(1 + r)(1 + r) = P(1 + r)2. Keep going, and after n periods:
A = P(1 + r)n
That is an exponential function of the period count n, with base 1 + r. Interest compounding on interest is what makes it exponential rather than linear. Take $1000 at 5% a year (so r = 0.05) for 10 years:
1000 · 1.0510 = $1628.89
Plain simple interest pays only on the original P each year: A = P(1 + r·n), a straight line. Compound interest pays on the running total: A = P(1 + r)n, a rising curve. Early on they nearly agree; over the years the curve pulls steadily above the line — the gap is "interest earning interest."
Run the same machine in reverse. Some quantities don't grow — they shrink by a fixed fraction each interval. A radioactive sample loses exactly half its atoms over a fixed span called the half-life T. If the amount starts at A0, then after time t there are t / T half-lives gone by, so:
A = A0·(½)t/T
This is just an exponential with a base between 0 and 1 — exponential decay. Carbon-14, used to date once-living things, has a half-life of about T ≈ 5730 years. A bone holding only 25% of its original 14C has lost half, then half again:
(½)1 = ½ → (½)2 = ¼ = 25%, so it is 2 half-lives ≈ 11 460 years old.
Growth (a > 1) rises; decay (0 < a < 1) falls toward the red asymptote y = 0. The model never reaches zero — there is always a sliver of 14C left — it just halves and halves and halves. That endless approach to y = 0 is the asymptote at work.
The natural follow-up question — "when does my money double? when does the sample reach 10%?" — puts the unknown right up in the exponent. You cannot reach it by the usual moves of add, subtract, multiply, divide; those touch the base, not the power. The tool built for exactly this job is the logarithm: take a log of both sides and the exponent drops down into the open.
Suppose $1000 at 5% is to double to $2000:
1000 · 1.05n = 2000 ⇒ 1.05n = 2 ⇒ n = log1.05 2 = lg 2lg 1.05 ≈ 14.2 years.
Reading "lg" as the base-10 log, lg 2 ≈ 0.3010 and lg 1.05 ≈ 0.0212, so n ≈ 0.3010 / 0.0212 ≈ 14.2. (The change-of-base formula from 24.3 Logarithms means any base works — ln 2 / ln 1.05 gives the same answer.)
For a fast mental estimate of doubling time, divide 72 by the percent rate. At 5%, 72 / 5 ≈ 14.4 years — within a whisker of the exact 14.2. At 6%, 72 / 6 = 12 years; at 8%, 72 / 8 = 9 years. The rule works because lg 2 / lg(1 + r) is close to 0.72 / r for small rates.
Bacteria double every hour: count = 2t (t in hours). When does the count first pass 100?
2t = 100 ⇒ t = log2 100 = lg 100lg 2 = 20.3010 ≈ 6.64 hours. The log brought the exponent down to where we can solve for it.
How do you draw a bacterium (about 10−6 m) and a blue whale (about 30 m) on the same axis? On an ordinary ruler the bacterium and the ant would be crushed into the first pixel. The fix is a log scale: instead of marking 1, 2, 3, …, you mark 100, 101, 102, … at equal spacing. Moving one step right no longer adds a fixed amount — it multiplies by ten.
On a log scale a value v sits at position log10 v. Equal distances mean equal ratios, not equal differences. That is why ten-fold jumps — a bacterium to a grain of sand, a grain to a person — all look like one tidy step. The whole point: a quantity spanning a dozen orders of magnitude fits comfortably on one line.
Three famous scales are built this way, because the things they measure span enormous ranges:
| Scale | Measures | One step up means |
|---|---|---|
| Richter | earthquake size | ~10× the ground amplitude (~32× the energy) |
| Decibel (dB) | loudness | +10 dB ≈ 10× the sound power |
| pH | acidity | one unit ≈ 10× the H+ concentration |
So a Richter 7 quake is not "a little bigger" than a Richter 5 — it shakes the ground about 107−5 = 102 = 100× as hard, and releases roughly 1000× the energy. The number 7 hides a thousandfold.
People badly underestimate exponentials. Take a sheet of paper 0.1 mm thick and imagine folding it in half over and over — each fold doubles the thickness, so after x folds it is 0.1 mm · 2x thick. Seven folds reach about a centimeter. Keep going. After 42 folds:
0.1 mm · 242 ≈ 4.4 × 108 m — past the Moon.
(You can't really fold paper 42 times — but the math is honest, and that is the point.) Now race the exponential against a line. Compare y = 2x to y = 100x. The line leaps ahead at first — at x = 5 it is at 500 while 25 is only 32. But the curve is gaining a multiple every step, and at x = 10 it overtakes for good: 210 = 1024 > 100·10 = 1000. After that the line never catches up again.
A line adds the same amount each step; an exponential multiplies by the same factor. Adding loses to multiplying in the long run, every time. No matter how steep the line — 100x, a million·x — a growing exponential (base > 1) will pass it and never be overtaken. That is why "exponential growth" is a phrase to respect.
One last thread. Powers and logarithms, like the lines and parabolas before them, only ever head one way — up or down — or turn once. But the world is full of change that comes back around: the swing of a pendulum, the daily tide, a turning wheel, an alternating current. Capturing repetition needs a new kind of function, one that cycles — the trigonometric functions of Stage 25. That is where the Functions strand heads next.
The exponential and the logarithm are working tools, not abstractions:
$2000 is invested at 3% per year, compounded annually. Write the formula for the balance after 5 years and find its value.
A = 2000·(1 + 0.03)5 = 2000·1.035 ≈ $2318.55. (Compound, not simple — simple interest would give only 2000·(1 + 0.15) = $2300.)
A wooden artifact retains 25% of its original 14C. How many half-lives have passed?
25% = ¼ = (½)2, so 2 half-lives have passed. (With T ≈ 5730 yr that is about 11 460 years old.)
A bacterial colony doubles every hour: count = 2n. Write "2n = 100" solved for n using a logarithm, and estimate it.
Take a log of both sides to free the exponent: n = log2 100 = lg 100 / lg 2 = 2 / 0.3010 ≈ 6.64 hours.
Use the Rule of 72 to estimate the doubling time of an investment earning 6% per year.
72 / 6 = 12 years. (The exact value lg 2 / lg 1.06 ≈ 11.9 years — the rule is a good quick guess.)
On the Richter scale, how much stronger is the ground shaking of a magnitude-8 earthquake than a magnitude-5 one?
Each unit is a factor of 10, and 8 − 5 = 3 units, so the amplitude is 103 = 1000× as large. (The released energy is even larger.)
For very large x, which is bigger: the line 100x or the exponential 2x? Explain in one sentence.
2x wins. The line adds a fixed amount each step while the exponential multiplies by 2; multiplying overtakes adding (here at x = 10, where 210 = 1024 > 1000) and never falls behind again.
Six questions to lock it in. Tap the answer you think is right.
This lesson is the payoff of Stage 24 — it shows learners why exponentials and logarithms earn their place in the curriculum. The single big idea: an exponential model, A = P(1 + r)n for growth or A = A0(½)t/T for decay, captures any process that changes by a fixed multiple each step; and whenever the unknown sits in the exponent, a logarithm is the only elementary tool that can reach it. The log scale is the same idea seen sideways — equal distances meaning equal ratios.
Misconceptions to watch. (1) Using simple interest where compound is meant — A = P(1 + rn) is a line; the compound model A = P(1 + r)n curves above it, and the difference grows with time. (2) Reading a log scale linearly — students think Richter 7 is "a bit more" than Richter 5, missing that the gap is a factor of 100 in amplitude (about 1000× in energy). (3) Believing a steep line eventually catches a slower-looking exponential — it never does; multiplying always wins over adding in the long run.
Common Core (high school): F-LE.A.1c and F-LE.A.2 (recognize and construct exponential growth/decay models), F-LE.B.5 (interpret the parameters P, r, T in context), F-LE.A.4 (solve a·bct = d for the exponent using logarithms), and F-IF.B.6 (rate of change / comparing growth). The log-scale and "exponential beats linear" threads also support modeling standards across F-IF and N-Q.