Tag: Exponential

Competing Curves

On By rolcottIn Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

It’s still October but there’s a distinct taste of oncoming November in the air — grey, gusty with a moist chill as I step into Cal’s coffee shop. “You’re looking a bit grumpy, Cal.”

“Sure am, Sy. Some lady come in here, wanted pumpkin spice. The nerve! I sell good honest high‑quality coffee, special beans and everything, no goofy flavors. You want peppermint or apple brown betty, go down to the mermaid place. Here’s your mugfull, double‑dark as always. By the way, fair warning — Richard Feder’s in town and looking for you. He’s at that corner table.”

“Thanks, Cal.” <sound of footsteps> “Morning, Mr Feder. How’d things go in Fort Lee?

“Nicely, nicely… I got a question, Moire.”

“Of course you do.”

“I been reading your stuff, you had a graph in one post looks just like the graph in a different post. Here, I printed ’em out. What’s up with that?”

“But they plot entirely different things, brightness against distance in one, atom loss against time in the other, completely different equations.”

“Yeah, yeah, but the shapes are the same I don’t care you say they got different equations. Look, they even both go through the same points at x=2 and 4. What’re you trying to pull here?”

“Not pulling anything. Those two curves are similar, yes, but they’re not identical.” <quickly building charts on Old Reliable> “Here, I’ve laid them both on the same axis. For good measure I’ve extended the x‑axis into a second panel with a stretched‑out y‑axis. What do you see?”

“Well, the orange one goes up and stops but it looks like the blue one’s headed for the sky.”

“It is. But where on the x-axis do those things happen?”

“Zero and one. Okay so the blue line squoze in a little.”

“How about out there at the x=8 end? Looks like they’re close, I’ll grant you, but check the y‑values at at the left of the second panel.”

“Uhh… Looks like blue’s four times higher than orange. Then the orange line flattens out but the blue line not so much.”

“Mm‑hm. So they behave differently at that end, too.”

“Yeah, but what about in the middle here” <jabs finger at Old Reliable’s screen> “where they’re real close and even cross over each other a couple times and you could just draw a straight line?”

“You’ve put your finger on something that challenges every theoretician and research experimentalist who works in a quantitative field. How do you connect the dots? Sure, you can eyeball a straight line through observed points sometimes, there are even statistical techniques for locating the best possible straight line, but is a straight line even appropriate? Sometimes it is, sometimes it’s not, and often we don’t know.”

“How can you not know? Everything starts with a straight line, shortest distance between two points, right?”

“Only if they’re the right points. Real observations are always uncertain. Lenses are never perfect, adjustment screws have a little bit of play, detector pixels are larger than a perfect point would be, whatever. Good experimentalists put enormous amounts of time and care into eliminating or at least controlling for every imaginable error source, but perfect measurements just don’t happen.”

“So it’ll be a fuzzy straight line.”

“For some range of ‘fuzzy’, mm‑hm. Now we get into the theory issues. We’ve already seen the simplest one — range of validity. Your straight‑line approximation might be good enough for some purposes in the x‑range between 2 and 4, but things get out of hand outside of that range.”

“Okay, in graphs. But these two curves both look good. Why choose one over the other?”

“That’s where theory and data collude. Sometimes theories tell us what data to look for, sometimes the data challenges us to develop an explanatory theory, sometimes we just try curve after curve until we find one that works across the full range that experiment can reach but we don’t know why. What’s exciting is when we get to use the data to determine which of several competing theories is the correct one. Or least incorrect.”

“I got other ways to get excited.”

“Of course you do.”

~~ Rich Olcott

The Oldest Clock Ticks Slowest

On By rolcottIn Astronomy: Techniques, Chemistry, UncategorizedLeave a comment

<Cliff‑hanger Cathleen strikes again> “How can you even measure a 2‑million year half‑life?”

Kareem’s right back at her. “What’s a half‑life?”

“Start a clock, weigh a sample, wait around for a while and then weigh it again to see how much is still there. When half of it’s gone, stop the clock and you’ve measured a half‑life. Simple.”

“Simple but not that simple or maybe a bit simpler. For one thing, you don’t have to wait for a full half‑life. For spontaneous radioactivity, all you have to know is the interval and whatever fraction disappeared. There’s a nice equation that ties those two to the half‑life.”

“Spontaneous? Like there’s another kind?”

“Stimulated radioactivity. That’s what nuclear reactors do — spew neutrons at uranium‑235 atoms, for instance, transmuting them to uranium‑236 so they’ll split into krypton and barium atoms and release energy and more neutrons. How often that happens depends on neutron concentration. Without that provoking push, the uranium nuclei would just split when they felt like it and that’s the natural half‑life.”

“Wait. I know that curve, it’s an exponential. Why isn’t e in the equation?”

“It could be. Would you prefer e-0.69315*t/half‑life? Works just as well but it’s clumsier. Base‑2 makes more sense when you’re talking halves. Usually when you say ‘exponential’ people visualize an increase. Here we’re looking at a decrease by a constant percentage rate but yeah, that’s an exponential, too. You get a falling curve like that from a Geiger counter and you’re watching counts per minute from someone’s thyroid that’s been treated with iodine‑131. Its 8‑day half‑life is slow enough to track that way. Really short half‑lives I don’t know much about; I care about the slow disintegraters that are either primordial or generated by some process.”

“Primordial — that means ‘back to the beginning’, which in your specialty would mean the beginning of the Solar System. We’re pretty sure the Sun’s pre‑planetary disk was built from dust broadcast by stars that went nova. Isotopes in the dust must be the primordial isotopes, right? Which ones are the other kind?”

“Mmm, aluminum‑26 is a good example. The half‑life equation still applies even for million‑year intervals. Half‑life of aluminum‑26 is about 0.7 million years and it decays to magnesium‑26. Whatever amount got here from the stars would have burnt down to a trillionth of that within the first 30 million years or so after arrival. Any aluminum‑26 we find today couldn’t be primordial. On the other hand, cosmic rays can smack a proton and neutron out of a silicon‑28 nucleus and voila! a new aluminum‑26. There’s a steady rain of cosmic rays out in space so there’s a steady production of aluminum‑26 out there. Not here on Earth, though, because our atmosphere blocks out most of the rays. Very nice for us geologists who can compare measured aluminum‑26 to excess magnesium‑26 to determine when a meteorite fell.”

“Excess?”

“Background magnesium is about 10% magnesium‑26 so we have subtract that to get the increment which came from aluminum‑26. A lot of the arguments in our field hinge on how much of which isotope is background or was background when a given rock formed. That’s one reason you see so much press about tiny but rugged zircons. They’re key to uranium‑lead dating. Crystallizing zirconium silicate doesn’t allow lead ions into its structure but it happily incorporates uranium ions. Uranium‑235 and uranium‑238 both decay to crystal‑trapped lead, but each isotope goes to a different lead isotope and with a different half‑life. The arithmetic’s simpler and the results are more definitive when you know that the initial lead content was zero.”

“So that aluminum‑magnesium trick’s not your only tool?”

“Hardly. The nuclear chemists have given us a long list of isotope chains, what decays to what with what half‑life and how much energy the radiating particle gets. Nuclei flit between quantum energy levels just like atoms and molecules do, except a spectrum of alpha or beta particles is a different game from the light‑wave spectrum. Tell me a radiated particle’s energy and I can probably tell you which isotope spat it out and disappeared.”

“Your ladder rungs are Cheshire Cat grins.”

~~ Rich Olcott

The Edge of Pinkness

On By rolcottIn Chemistry, Mathematics: Logs and Exponentials, Uncategorized1 Comment

Susan Kim takes a sip of her mocha latte. eyes me over the rim. “That’s quite a set of patterns you’ve gathered together, Sy, but you’ve left out a few important ones.”

“Patterns?”

A log-linear plot

“Regularities we’ve discovered in Nature. You’ve written about linear and exponential growth, the Logistic Curve that describes density‑limited growth, sine waves that wobble up and down, maybe a couple of others down‑stack, but Chemistry has a couple I haven’t seen featured in your blog.”

“Such as?”

“Log-linear relationships are a biggie. We techies use them a lot to handle phenomena with a wide range. Rather than write 1,000,000,000 or 109, we sometimes just write 9, the base‑10 logarithm. The pH scale for acid concentration is my favorite example. It goes from one mole per liter down to ten micro‑nanomoles per liter. That’s 100 to 10-14. We just drop the minus sign and use numbers between 0 and 14. Fifteen powers of ten. Does Physics have any measurements that cover a range like that?”

“A handful, maybe, in theory. The limitation is in confirming the theory across a billion-fold range or wider. Atomic clocks that are good down to the nanosecond are our standards for precision, but they aren’t set up to count years. Mmmm … the Stefan‑Boltzmann Law that links an object’s electromagnetic radiation curve to its temperature — our measurements cover maybe six or seven powers of ten and that’s considered pretty good.”

“Pikers.” <but I like the way she grins when she says it>

“I took those Chemistry labs long ago. All I remember was acids were colorless and bases were pink. Or maybe the other way around.”

“You’ve got it right for the classic phenolphthalein indicator, but there are dozens of other indicators that have different colors at different acidities. I’ll tell you a secret — phenolphthalein doesn’t kick over right at pH 7, the neutral point. It doesn’t turn pink until the solution’s about ten times less acidic, near pH 8.”

Adapted from this file by Damitr, CC BY-SA 4.0

“So all my titrations were off by a factor of ten?”

“Oh, no, that’s not how it works. I’m going to use round numbers here, and I’ll skip a couple of things like the distinction between concentration and activity. Student lab exercises generally use acid and base concentrations on the order of one molar. For most organic acids, that’d give a starting pH near 1 or 2, way over on the sour side. In your titration you’d add base, drop by drop, until the indicator flips color. At that point you conclude the amounts of acid and base are equivalent, not by weight but by moles. If you know the base concentration you can calculate the acid.”

“That’s about what I recall, right.”

“Now consider that last drop. One drop is about 50 microliters. With a one‑molar base solution, that drop holds 50 nanomoles. OK?”

<I scribble on a paper napkin> “Mm-hm, that looks right.”

“Suppose there’s about 50 milliliters of solution in the flask. Because we’re considering the last drop, the solution in the flask must have become nearly neutral, say pH 6. That means the un‑neutralized acid concentration was 10-6 moles per liter, or one micromolar. Fifty milliliters at one micromolar concentration is, guess what, 50 nanomoles. Your final drop neutralizes the last of the acid sample.”

“So the acid concentration goes to zero?”

“Water’s not that cooperative. Water molecules themselves act like acids and bases. An H2O molecule can snag a hydrogen from another H2O giving an H3O+ and an OH–. Doesn’t happen often, but with 55½ moles of water per liter and 6×1023 molecules per mole there’s always a few of those guys hanging around. Neutral water runs 10-7 moles per liter of each, which is why neutral pH is 7. Better yet, the product of H3O+ and OH– concentrations is always 10-14 so if you find one you can calculate the other. Take our titration for example. One additional drop adds 50 nanomoles more base. In 50 milliliters of solution that’s roughly 10-6+10-7 molar OH–. Call it 1.1×10-6, which implies 0.9×10-8 molar H3O+. Log of that and drop the minus sign, you’re a bit beyond pH 8 which sends phenolphthalein into the pink side. Your titration’s good.”

I eye her over my mug of black mud. “A gratifying indication.”

~~ Rich Olcott

Something of Interest

On By rolcottIn Mathematics: Logs and Exponentials, Physics: Money, UncategorizedLeave a comment

“OK, Sy, I get how money is sorta like Physics ‘energy‘ except you can’t create energy but you can create money. And I get how Economics ‘velocity of money‘ and Physics ‘velocity‘ don’t have much to do with each other. Your ‘Money Physics‘ phrase doesn’t make much sense unless you’ve got something with more overlap than that.”

“You’re a tough man, Vinnie. How about the word ‘exponential‘?”

“Means something goes up really fast. What about it?”

“Well, first off that’s not really what it means and that’s one of my personal peeves, thank you very much. Yes, quantities can increase exponentially, but not necessarily rapidly, and they can also decrease exponentially, either fast or slow. It’s a math thing.”

“Alright, I got myself into this. You’re gonna tell me how that works and it probably involves equations.”

“You made the phone call, I’m just sitting here, but you’re good, no equations just arithmetic. Ten times ten’s a hundred, right, and you can write that either 10×10 or 10², OK? The little two is the exponent, tells you how many factors to multiply together.”

“And 10 with a little three makes a thousand and ten with a little … six makes a million. See, it goes up really fast.”

“Depends on what the base number is. I’ve sent a tabulation to your phone…”

Exp’t 10 2 99% 100% 101%
2 100 4 98.01% 100% 102.01%
3 1 000 8 97.03% 100% 103.03%
4 10 000 16 96.06% 100% 104.06%
5 100 000 32 95.10% 100% 105.10%
6 1 000 000 64 94.15% 100% 106.15%
7 10 000 000 128 93.21% 100% 107.21%

“What’s all that?”

“Well, the top-row headers are just numbers I multiplied by themselves according to some exponents, and the first column is the series of exponents I used. Like we said, 10² is a hundred and so on down the second column. Number 2 multiplied by itself according to the same exponents gave me the third column and you see the products don’t grow anywhere near as fast. Do you see how the growth rate depends on the number that’s being multiplied and re‑multiplied?”

“No problem. What about the other columns?”

“Start with the fifth column. What’s 100% of 100%?”

“All of it.”

“And 100% of 100% of 100%?”

“I get it — no change no matter the exponent.”

“Absolutely. Now compare that to the 99% and 101% columns that give you the effect of a 1% growth factor. As you’d expect, very little change in either one, but there’s a lesson in the 99% column. It’s exponential by definition, but the results go down, not up. By the way, both of those are such small factors that the results are practically linear. You need to get beyond 15% factors for visible curvature in the usual graphs.”

“OK, so exponential says some arithmetic factor gets applied again and again. What’s that got to do with Physics or Economics?”

“Ever since Newton, Physics has been the study of change, all different kinds. Gradually we’ve built up a catalog of change patterns. Newton pointed out the simplest one in his first Law of Motion — constant velocity, say in meters per second. Plot cumulative distance moved against time and you get a rising straight line. His Second Law implies another simple pattern, constant acceleration. That’s one where velocity’s line rises linearly but distance goes up as the square of the time traveled. But Newton never tackled another very simple, very common pattern.”

“I thought Newton did everything.”

“Not the case. He was an amazing geometer, but to handle this pattern you need algebraic tools like the ones Leibniz was developing. Newton would rather have dunked his arm in boiling rancid skunk oil than do that. It took another century or so until the Bernoulis and Euler beat that problem into the ground.”

“So what’s the simple pattern?”

“Suppose instead of a quantity increasing by some absolute number of thingies per second, it increases by some constant percentage. That’s uncommon in the kinds of mechanical phenomena that Newton studied but it does happen. Say you’re a baby planet in the middle of a dust cloud. Get 15% bigger, you’re 15% better at attracting even more dust. Biological things do that a lot — the more bugs or bacteria you’ve got, the faster they multiply and that’s usually at a constant percentage-per-time rate. Exponential growth in a nutshell.”

“Planets, bugs, what’s that got to do with Economics?”

“Ever hear of ‘compound interest‘?”

“Low rates on bank accounts, high rates on credit cards, compounded. Gotcha.”

“Inflation does compounding, too.”

~~ Rich Olcott

The Curve To Be Flattened

On By rolcottIn Mathematics: Logs and Exponentials, UncategorizedLeave a comment

<chirp chirp> My phone’s ring-tone for an non-business call. “Moire here.”

“Mr Moire, it’s Jeremy.”

“I hope so, Jeremy, my phone shows your caller-ID. I’m glad you called instead of trying to drop by, the city being under lockdown orders and all. What’s your question?”

“Oh, no question, sir, I just called to chat. It’s lonely over here. If you’ve got the time, anything you’d like to talk about would be fine.”

“Mm… Well, I am working on a project but maybe talking it out will help get my thoughts in order. Have you seen that ‘Flatten the curve‘ chart?”

“Sure, it’s been hard to escape. They use it to tell us why we shouldn’t do group stuff while this virus is going around. Are you writing about where the chart comes from?”

“That’s my project, all right. There’re two ways to get to that chart and I’m trying to decide which will work better. I could start from ecology studies of invading organisms taking over a new territory. At first the organisms multiply rapidly, doubling then doubling again —”

“That’s exponential growth, Mr Moire. We talked about that!”

“Just sent you an image. When researchers plot invasions they usually look like the black line, the Logistic Curve. Its height represents the organism’s population as time increases left-to-right. At the beginning there’s that exponential rise. Over on the right the growth rate slows as the plants or animals or bugs use up increasingly scarce resources. The part in the middle’s almost linear. All that’s a familiar story by now, right?”

The Logistic Curve (black) and its slope (red)

“Uh-huh. We talked a lot about ecology back in kid school except we hadn’t learned graphs yet. What’s the red curve?”

“That’s the interesting part I’m trying to write about. One way to look at it is that it’s simply the slope of the Logistic curve. See how where the Logistic is rising, the slope is rising, too? That’s the way exponentials work — ‘the higher the faster‘ as they say. The slope switches direction just where the Logistic switches from growth to slow-down. The Logistic Curve approaches its limit when the organism’s population approaches the carrying capacity of the territory. That’s also where the slope gets shallowest. Very few resources, very little expansion.”

“What’s the other way to look at it?”

“We start with the slope curve itself. It has its own straight-forward interpretation, especially if the organism is a a bacterium or virus that causes disease. Consider the population under attack as the resource. How fast will the disease spread?”

“Uh… what I keep hearing is that if more people get sick, other people will get infected faster.”

“But what happens when nearly everyone’s caught it and they’ve either recovered or left us?”

“Oh, there’ll be fewer people left to catch it so the disease spreads more slowly.”

“Let me put that into algebra. I’ll write N for the total number of people and that’ll be a constant, we hope. At any given time we’ve got S as the current number of people who are susceptible. Then (N‑S) tells us how many people are NOT susceptible. Are you with me?”

“Fine so far.”

“So from what we’ve just said, the rate of infection is low when S is low and also low when (N‑S) is low. One way to make that into an equation is to write the rate as R = K*S*(N‑S). K is just a number we can adjust to account for things like virulence and Social Distance effectiveness. If we plot R against time what shape will it have?”

“Mmm… S is nearly the same as N at the start so (N‑S) is nearly zero then. At the finish, S is nearly zero. Exactly in the middle S equals (N‑S). They each have to be higher than near-zero there. That makes R be low at each end and high in the middle. Ah, that’s sort-of the shape of the slope curve!”

“It’s exactly the shape of the slope curve. So how do we flatten it?”

<click-click, click-click> “Oops, Mr Moire, my phone battery’s about dead. Gotta go get the charger. I’ll be right back.”

“I’ll be here, Jeremy.”

~~ Rich Olcott

A log by any other name

On By rolcottIn Mathematics: Logs and Exponentials, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

“Hey, Mr Moire?”

“Yes, Jeremy?”

“What we did with logarithms and exponents.  You showed me how my Dad’s slide-rule uses powers of 10, but we did that compound interest stuff with powers of 1.1.  Does that mean we could make a slide-rule based on powers of any number?”

“Sure could, in principle, but it’d be a lot harder to use.  A powers-of-ten model works well with scientific notation.  Suppose you want to calculate the number of atoms in 5.3 grams of carbon.  Remember Avagadro’s number?”

“Ohhh, yeah, chem class etched that into my brain.  It’s 6.02×10²³ atoms per gram atomic weight.  Carbon’s atomic weight is 12, so the atom count would be (5.3 grams)×(6.02×10²³ atoms / 12 grams), whatever that works out to be.”

“Nicely set up.  With the slide-rule you’d do the 5.3×6.02/12 part, then take care of the ten-powers in your head or on a scrap of paper.  It’d be ugly to do that with a slide-rule based on powers of π, for example.  Although, once you get away from the slide-rule it’s perfectly possible to do log-and-exponent calculations on other bases.  A couple of them are real popular.  Base-2, for instance.”2-10-e logs

“Powers of two?  Oh, binary!   2, 4, 8, 16, like that.  And 1/2, 1/4, 1/8.  Hard to imagine what a base-2 slide-rule would look like — zero at one end, I suppose, and one at the other and lots of fractions in-between.”

“Well, no.  Is there a zero on your Dad’s base-10 slide-rule there?”

“Uh, no, the C scale has a one at each end.”

“The left-hand ‘1’ can stand for one or ten or a thousand or a thousandth.  Whatever you pick for it, the right-hand ‘1’ stands for ten times that.”

“Ah, then a base-2 slide-rule would also have ones at either end in binary but they’d mean numbers that differ by a factor of two.  But there’d still be a bunch of fractions in-between, right?”

“Right, but no zero anywhere.  Why not?”

“Oh, there’s no power-of-two that equals zero.”

“No power-of-anything that equals zero.  Except zero, of course, but zero-to-anything is still zero so that’s not much use for calculating.  On the other hand, anything to the zero power is 1 so log(1)=0 in every base system.”

“You said a couple of popular bases.  What’s the other one?”

“Euler’s number e=2.71828…  It’s actually closely related to that compound interest calculation you did.  There’s several ways to compute e, but the most relevant for us is the limit of [1+(1/n)]n as n gets very large.  Try that on your spreadsheet app.”

“OK, I’m loading B1 with =(1+(1/C1))^C1 and I’ll try different numbers in C1.  One hundred gives me 2.7048, a thousand gives me 2.7169 (diminishing returns, hey) — ah, a million sure enough comes up with 2.71828.”

“There you go.  Changing C1 to even bigger values would get you even closer to e‘s exact value but it’s one of those irrationals like π so you can only get better and better approximations.  You see the connection between that formula and the $×[1+(rate/n)]n formula?”

“Sure, but what use is it?  If that’s the e formula the rate is 100%.”

“You can think of e as what happens when growth is compounded continuously.  It’s not often used in retail financial applications, but it’s everywhere in advanced math and physics.  I don’t want to get too much into that because calculus, but here’s one specialness.  The exponential function ex is the only one whose slope at every point is equal to its value there.”

“Nice.  But we’ve been talking logs.  Are base-e logarithms special?”

“So special that they’ve got their own name — natural logarithms, as opposed to common logarithms, the base-10 kind that power slide-rules.  They’ve even got their own abbreviations — ln(x) or loge(x) as opposed to log(x) or log10(x).”

“What makes them ‘natural’?”

“That’s harder to answer.  The simplest way is to point out that you can convert a log on one base to any other base.  For instance, ln(10)=2.303 therefore e2.303=10=101.  So log10 of any number x is 2.303 times ln(x) and ln(x)=log10(x)/2.303.  There are loads of equations that look simple and neat in terms of ln but get clumsy if you have to plug in 2.303 everywhere.”

“Don’t want to be clumsy.”

~~ Rich Olcott

Powers to The People

On By rolcottIn Mathematics: Logs and Exponentials, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

“You say logarithms and exponents have to do with growth, Mr Moire?”Log Exp and slide rule captioned

“Mm-hm.  Did they teach you about compound interest in that Modern Living class, Jeremy?”

“Yessir.  Like if I took out a loan of say $10,000 at 10% interest, I’d owe $11,000 at the end of the first year and, um…, $12,100 after two years because the 10% applies to the interest, too.”

“Nice mental arithmetic.  So what you did was multiply that base amount by 1+10% the first year and (1+10%)² the second, right?”

“Well, that’s not the way I thought of it, but that’s the way it works out, alright.”

“So it’d be (1+10%)³ the third year and in general (1+rate)n after n years, assuming you don’t make any payments.”

“Sure.”

“OK, how do we have to revise that formula if the interest is compounded daily and you get lucky and pay it off in a lump sum after 19 months?”

“Can I use your whiteboard?”

“Go ahead.”

“OK, first thing to change is the rate, because the 10% was for the whole year.  We need to use 10%/365 inside those parentheses.  But then we’re counting time by days instead of years.  Each day we multiply the previous amount by another (1+10%/365), which makes the exponent be the number of days the loan is out, which is 19 times whatever the average number of days in a month is.”

“Why not just use 19×(365÷12)?”

“Can we do that?  In an exponent?”

“Perfectly legal, done in all the best circles.”

“So what we’ve got is
10000×[1+(10%/365)]19×(365÷12).”

“Try poking that into your smartphone’s spreadsheet app and format it for dollars.”

“In spreadsheet-ese that’d be
=10000*(1+(0.1/365))^(19*(365/12)).
Hah!  The app took it, and comes up with … $11,715.31.  Lemme try that with two years that’s 24 months.  Now it’s $12,213.69.  Hey, that’s $123 more than two years compounded once-a-year.  Compounding more often generates more interest, doesn’t it?”

“Which is why daily compounding is the general rule in consumer lending.  But there’s a couple more lessons to be learned here.  One, you can do full-on arithmetic inside an exponent.  That’s what the log log scales are for on a slide rule.  Two, the expression you worked up has the form
base×(growth factor)(time function).
Any time you’re modeling something that grows or shrinks in some percentage-wise fashion, you’re going to have exponential expressions like that.”

“Hey, I tried compounding more often and it didn’t make much difference.  I put in 3650 instead of 365 and it only added 30¢ to the total.”

“Which gives me an idea.  Load up cells A1:A7 in your spreadsheet with this series: 1, 3, 10, 30, 100, 300, 1000.  Got it?”

“Ahhh … OK.  Now what?”

“Now load cell B1 with +10000*(1+(0.1/A1))^(24*(A1/12)).”

“Says $12,100.”

“Fine.  Now copy that cell down through B7.”

“Hmm…  The answers go up but by less and less.”

“Right.  Now highlight A1:B7 and tell your spreadsheet to generate a scatter plot connected by straight lines.”

“Gimme a sec … OK.  The line goes straight up, then straight across almost.”

“Final step — click on the x-axis and tell the program to use a logarithmic scale.”diminishing returns

“Hey, the x-numbers scrunch and wrap like on the A, B and K scales on Dad’s slide-rule.”

“Which is what you’d expect, right?  They both use logarithmic scales.  The slide-rule uses logarithms to do its arithmetic thing.  The graphing software lets you use logarithms to display big numbers together with small numbers.  But the neat thing about this graph is that it shows two different flavors of a general pattern.  Adding something, say 20, to a number to the left on the x-axis moves you a longer distance than adding the same amount somewhere over on the right.  That’s diminishing returns.”

“Look, the heeling-over curve shows diminishing returns from compounding interest more and more often.”

Exploding returns“Good.  Now copy A1:A7 by value into C1:C7 and generate a scatter plot of B1:C7. This time apply the logarithmic scale to the y-axis. This’ll show us how often we’d need to compound to get the yield on the x-axis.”

“Whoa, it blows up, like there’s no way to get up to $12,300.”

“Call it exploding returns.  Increasing the exponent increases the growth factor’s impact.  Beyond a threshold, a small change in the growth factor can make a huge difference in the result.”

“Seriously huge.”

“Exponentially huge.”

~~ Rich Olcott