What does an exponent really mean?

Ever get a hint of confusion about what an exponent was doing? I sure have.

Like the word "run", the meaning depends on context:

  • crawl / walk / run (movement)
  • run a company (general operation)
  • a run of good luck (sequence)
  • and a dozen more definitions

Sticking with a single interpretation of "run" leads to confusion, and the same happens in math. Let's clarify how exponents are used.

What does an exponent really mean?

Meaning 1: Repeated Multiplication

We first learn that exponents like $3^2$ or $a^n$ are repeated multiplication: multiply $a$, $n$ times.

Like counting on your fingers, this breaks down beyond the positive integers. What does a fractional exponent mean? A negative one? Zero? (Since $a^0 = 1$, we multiply zero times and get 1?)

Common usage of $a^n$: Counting problems. If you flip a coin $n$ times, you have $2^n$ possible outcomes.

Meaning 2: Growth Microwave

Let's say I have an exponent like $3^{4.5}$. I mentally convert it to $1.0 * 3^{4.5}$, and then $1.0 * g^{t}$.

With the "growth microwave" analogy, an exponent grows our starting amount (1.0) by $g$ for $t$ units of time. (In this example, 3x growth applied for for 4.5 seconds.)

What values can $t$ have?

  • If t is positive, we go forward in time and get larger (assuming $g > 1$). Fractional time is ok -- I can run a microwave for 3.5 minutes, and get some effect between 3 minutes and 4 minutes.
  • If t is negative, we go backwards in time and get smaller. If a regular microwave allowed negative time, it would cool down your food, right?
  • If t is zero, we didn't use the machine at all! We're left with 1.0, our original amount.

The growth microwave interpretation helps with fractional powers (and resolves the t=0 issue), but it's not flexible. Doubling the rate and halving the time doesn't have the result we expect:

\displaystyle{3^2 \neq 6^1}

2 seconds of 3x growth isn't the same as 1 second of 6x growth. Ugh. I'm not a caveman, we need to mix rate and time! (Hold onto that thought.)

Common usage of $g^t$: Man-made systems. If I agree to pay you 15% at a certain discrete interval (yearly), we can model the outcome as $(1.15)^t$. If I decide to cut the payments short (2.5 years) we can exponentially interpolate between the two intervals ($r^{0.5}$ is the square root). We often set $g = (1 + r)$, so we could write $(1 + .15)^t$.

Aside: Let's prove $g^t$ isn't flexible.


\begin{aligned}
g^t &\stackrel{?}{=} (gn)^{t/n}  \ [\text{claim to test}] \\
g^{tn} &\stackrel{?}{=} (gn)^{t} \ [\text{raise both sides to nth power}] \\
g^{n} &\stackrel{?}{=} (gn) \ [\text{raise both sides to (1/t) power}] \\
g^{n} &\neq (gn) \ [\text{not true in general}]
\end{aligned}

However, this shows the special case of $2^2 = 4^1$ does work.

Meaning 3: Continuous Growth Engine

Regular readers know I think of e as a continuous growth engine:

Instead of waiting to grow at discrete intervals, we apply interest immediately and compound as fast as we can. A pleasant consequence of e's definition is that we merge rate and time into a single, interchangeable quantity:

\displaystyle{e^{50\% \cdot 2} = e^{100\% \cdot 1} = e^{r \cdot t} = e^{x}}

Conveniently, 2 years of 50% growth is the same as 1 year of 100% growth. We doubled our rate, halved our time, and got the same result. (Practically, we may prefer the shorter time period but the final quantity is the same.)

The input $x$ is the "growth fuel" that can be separated into "rate * time". The base, e, is a machine that just cares how much fuel you gave it. Drip the fuel over 50 time periods, or firehose it into a single one. Either way, the same total input $x$ gets the same final result.

Common usage of $e^{rt}$: Natural systems. Most laws of physics have continuous growth patterns (no delay between earning interest and using it). We may occasionally use the man-made version for our convenience, e.g., describing a radioactive half life of 20 years, even though the atoms are decaying on an instant-by-instant basis.

(Aside: Use the natural log to convert one exponent format to another. $g^t = e^{\ln(g)t}$)

Meaning 4: A Power Series Calculation

We can treat $e^x$ as a fancy mathematical function:

\displaystyle{e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...}

You may see $e^x$ written $\exp(x)$, treated like any other function $f(x)$. Here, $x$ is just a numerical input to an intricate power series. Concepts like repeated counting, growth rate, and time fall into the background (though we can see them if we look).

Curiously, we're left with integer powers ($x^0, x^1, x^2, x^3$) and our "repeated multiplication" interpretation shows up again! The power of the exponent, $x$, switches from the number of multiplications to the base being multiplied. (The ciiiircle of life.)

Common usage of $\exp(x)$: When we see $e^x$ as just another function, a few properties emerge:

  • Using calculus with exponents gets way easier, since we can take the derivative / integral of each term (and realize $\frac{d}{dx} e^x = e^x$).
  • Exponential approximations become easy: $e^x \sim 1 + x$ for small values of $x$, since the higher-order powers become negligible.
  • Other math patterns click. Sine and cosine have expansions similar to $e^x$, hinting that trig functions and exponents are connected (Euler's Formula).
  • $e^x$ looks like a polynomial of infinite degree, and will eventually surpass any finite polynomial. (While $x^2 + 100 > e^x$ in the beginning, $e^x$ will eventually exceed it.)

So which version of exponents is best?

You probably guessed it: it depends, though the interpretations are listed from most to least common for a general audience.

If a formula doesn't make sense, try switching versions. Life's too short to have only a single interpretation of exponents.

Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Q: Why is e special? (2.718…, not 2, 3.7 or another number?)

A common question is why e (2.71828...) is so special. Why not 2, 3.7 or some other number as the base of growth?

First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't originally decided to be 299,792,458 m/s -- we did experiments and realized under ideal, universal conditions (a vacuum), this was the fastest light could move1.

Let's look at growth and ask under ideal, universal conditions, what's the fastest something can possibly grow? Ideal, universal assumptions would be:

  • Growing by the unit rate (100%)
  • Growing for the unit time (1 time period)
  • Growing perfectly, without any delay (continuous)

Turning these assumptions into a formula, we get:

If we actually use the formula (using large values of n for more accuracy) we get e = 2.718281828459...

Objection: But $13.74^x$ can model exponential growth just like $e^x$ can!

Sure. But what assumptions did you make to get 13.74? They probably weren't "unit rate, unit time, perfectly compounded". (You can pick k as the speed of light through Kool Aid too -- but why?)

Arguably, $2^n$ is also universal ("the discrete e"), because you have zero compounding (n is an integer like 0, 1, 2, 3). Instead of perfectly continuous, it's perfectly non-continuous (discrete), and we take growth step-by-step.

So, I'd say either $2^n$ (in discrete systems) or $e^x$ (in continuous systems) are "universal".

Objection: But things can grow faster than $e^x$, which is just $2.71828^x$ -- what about $13.74^x$?

What is it with you and 13.74? Yes, you can beat $e^x$ in an exponential footrace, if you use a rate more than 100%. $13.74^x$ is really $[e^{\ln(13.74)}]^x$. Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% $e^x$ uses. (Alternatively, you can grow for 260% of the unit time period that $e^x$ uses.)

Related:


  1. Funny enough, in 1983 c was decreed to be 299,792,458 m/s by redefining the length of a meter. Similarly, you could decide that e is a clean "10" in your base-e number system. 

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Common Definitions of e (Colorized)

Seeing the same math concept from a few perspectives helps build intuition. Seeing that e is my favorite constant (sorry, pi), a while back I put the definitions of e together to visualize their connection:

alternative explanations of e

The key intuition: e represents 100% continuous growth.

Today let's revisit each definition with a colorization viewpoint, describing continuous growth from a few different perspectives.

Definition 1: Compound Interest Perfectly

e colorized equation

This definition of e was my starting point on understanding the concept. We start with 1 growing to 2 (100% interest), and then compound that growth more and more frequently.

Eventually, we see that 1 grows to 2.71828..., hitting a speed limit of e.

The trick is distinguishing the role of each "1" in the definition. One is the base quantity, one is the interest, and another is the implicit single unit of time we plan to grow for. Math is so abstract that we don't have these separations labeled individually, they are just quantities interacting.

Definition 2: Track Each Interest Contribution

e definition colorized series

This definition splits up the compounding process into chunks we can see separately. I like to see each component like a "factory" that is earning money. We start with our initial amount, which builds interest. That interest builds its own interest, whcih builds its own interest, and so on (read more).

From a calculus perpsective, here's what's happening:

  • Our initial quantity is 1 (for all time)
  • This principal earns 100% continuous interest, and after time $x$ has earned $\int 1\, dx = x $. (After 1 unit of time, this is 1)
  • After time x, that interest ($x$) has earned $\int x \,dx = \frac{1}{2} x^2$. (After 1 unit of time, this is $\frac{1}{2}$)
  • After time x, that interest ($\frac{1}{2}x^2$) has earned $\int \frac{1}{2} x^2 \,dx = \frac{1}{2}\frac{1}{3}x^3 = \frac{1}{3!}x^3$ (After 1 unit of time, this is $\frac{1}{3!} = \frac{1}{6}$)

And so on. Every instant, the entire chain of interest is growing. When learning calculus, you might have repeatedly tried to integrate $x$ just for fun (whatever gets you going). That game is how we end up with $e$.

Instead of plugging in $x=1$ to compute $e^1$, we can leave $x$ unspecified to handle any amount of time:

\displaystyle{e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots }

I like seeing how each piece of interest contributes to the whole. Later terms have larger powers, but fight a larger factorial. For very small values of $x$ (like 1%), we can approximate $e^x$ as:

\displaystyle{e^x \sim 1 + x}

For example, earning 1% continuous interest for a single year is still around 1.01, because there isn't much growth from compounding. (And yep, $e^.01 = 1.01005$.)

Definition 3: Maintain 100% Continuous Growth

e colorized definition derivative

This is the shortest definition, but relies on calculus machinery. In short, we're saying that $e^x$ always changes by exactly the amount that we have.

The derivative ($\frac{d}{dx}$) measures instantaneous change:

And we say $e^x$ is that input that makes this machine return the original value. (In a similar way, "0" is the input that makes addition return the original; 1.0 is the input that makes multiplication return the original value.)

We can check this works. We saw earlier that $e^x$ is really this equation:

\displaystyle{e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots }

If you take the derivative each term in the right-hand side we get:

\displaystyle{e^x = 0 + 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + \dots }

which simplifies to

\displaystyle{e^x = 1 + x + \frac{x^2}{2!} + \dots }

In other words, every term gets "pulled over" to the left, with the constant 1.0 disappearing (it doesn't change). We have our original pattern, therefore $e^x$ is its own derivative. (There's no +C chicanery here, because $\frac{d}{dx} (e^x + C) \neq e^x + C$).

While other functions like $f(x) = x^2$ or $f(x) = \sin(x)$ may momentarily equal their derivative at certain instants, they don't keep it up for all values of $x$. $e^x$ is that Disneyland ride that keeps the magic going forever.

Definition 4: Define e using the natural log

natural log colorized equation definition

This definition is the most gnarly: instead of talking about e directly, we work backwards.

Define the natural logarithm as the time needed to grow from 1 to $a$ (our desired number), assuming 100% continuous interest. What does that look like?

Let's say we've grown to 4.0. How long to grow to 5.0? Well, assuming 100% interest, we grow 4 units per time period, so it takes us 1/4 of a unit to grow to 5.

And when we're at 5, it'll take us 1/5 of a unit to get to 6.

And so on. The time to grow from to 1 to a is the time from 1 to 2 (1 unit), plus 2 to 3 (1/2 unit), plus 3 to 4 (1/3 unit), until we reach $a$.

(In reality, we need to break time down microscopically, growing from 1, to 1.1, to 1.2, etc. That's what the integral $\int\frac{1}{x}\,dx$ really does.)

Phew! Once we have the notion of "time to grow from 1 to a" defined, we say e is the number that takes 1 unit of time to reach. In other words:

\displaystyle{\int_1^e \frac{1}{x}dx = 1}

Here, e is the "base of the natural logarithm".

Trace the similarities

I feel comfortable with an idea when I can hop between definitions and notice similarities. For example, look at the items that show up in each colorization: do you see where "interest" shows up in each definition? The unit quantity? The idea of perfection or infinitely precise change?

It feels great when e becomes a flexible tool on your bat-belt and not an incantation to memorize.

Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Understanding Discrete vs. Continuous Growth

There are two types of exponential growth, and it's easy to mix them up:

  • Discrete growth: change happens at specific intervals
  • Continuous growth: change happens at every instant

Here's the difference:

discrete vs continuous growth diagram

The key question: When does growth happen?

With discrete growth, we can see change happening after a specific event. We flip a coin and get new possibilities. We have a yearly interest payment. A mating season finishes and offspring are born.

With continuous growth, change is always happening. We can't point to an event and say "It changed here". The pattern is always in motion (radioactive decay, a bacteria colony, or perfectly compounded interest).

(Brush up on the number e and the natural logarithm.)

Insight: Convert between discrete and continuous

I visualize change as events along a timeline:

large discrete changes vs. small continuous ones

Discrete changes happen as distinct green blobs. We can take them, split them into smaller, more frequent changes, and spread them out. With enough splits, we could have smooth, continuous change.

So, discrete changes can be modeled by some equivalent, smooth curve. What does it look like?

discrete to continuous

The natural log finds the continuous rate behind a result. In our case, we grew from 1 to 2, which means our continuous growth rate was ln(2/1) = .693 = 69.3%. The natural log works on the ratio between the new and old value: $\frac{\text{new}}{\text{old}}$.

Mathematically,

\displaystyle{
2^x = e^{\ln(2)x} = e^{.693 x}
}

In other words: 100% discrete growth (doubling every period) has the same effect as 69.3% continuous growth. (Continuous growth requires a smaller rate because of compounding.)

So which version do we use?

Now here's the question: how should we talk about growth? It depends on the scenario:

  • If growth happens in a man-made system, discrete growth works better ($2^x$, $3^x$)
  • If growth occurs a natural system, continuous growth is better ($e^x$)

Let's take a look.

Example: Flipping Coins

Let's say we flip a coin. What are the possible outcomes?

  • 1 flip: 2 outcomes (H or T)
  • 2 flips: 4 outcomes (HH, HT, TH, TT)
  • 3 flips: 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)

You see where this is going. I'd describe the number of possibilities as $2^n$ where n was the number of flips.

I'm using "n" (not x) by convention: x could mean any value on the x-axis (-3, 1.234, $\sqrt{14}$), while n represents an integer (1, 2, 3, 4).

Could we say the number of outcomes was $e^{\ln(2)x}$, where x was the number of coin flips? Yes. But it's confusing: in a man-made system, where we have change events, I'd use the discrete version to describe the possibilities.

Example: Binary Numbers

Binary numbers follow the same pattern: if we have n bits, we get $2^n$ possibilities. For example, 8 bits have 256 possible values, and 16 bits have 65536.

(There may be some cases where intermediate values make sense, like representing the number of bits required, even though we need a whole number of bits in practice. This is similar to saying the average family has 2.3 kids.)

Example: Radioactive Decay

When radioactive material decays, we often talk about its half-life: how long until half the material is gone?

For example, the half-life of Carbon-14 is 5700 years. We could write it like this:

\displaystyle{
\text{percent of carbon left} = (1/2)^{\text{years}/5700}
}

If we wait 5700 years, we expect $(1/2)^1= .50$ of the carbon remaining. If we double that and wait 11,400 years, we'd expect $(1/2)^2 = .25$ of the carbon left.

However, this equation is written for our convenience. Carbon doesn't decay in jumps, politely waiting around 5700 years and suddenly decaying by half. We use (1/2) as the base because we humans want to count the number of halvings (decaying into half, decaying into a quarter, decaying into an eighth...).

The radioactive material is changing every instant. From a physics perspective, a continuous rate is more telling. We can find the continuous decay rate by converting the discrete growth into a continuous pattern:

\displaystyle{
(1/2)^{\text{years}/5700} = (e^{\ln(1/2)})^{\text{years} / 5700} = e^{-.693 \cdot \text{years} / 5700} = e^{-0.00012 \cdot \text{years}}
}

This helps me understand why the natural log is natural -- it's describing what nature is doing on an instant-by-instant basis. None of this "wait until we decay by 50% so humans can count it easier" nonsense.

In practice, you don't discover the half-life by waiting for carbon to decay 50%. You'd wait a reasonable about of time (a year?), use the natural log to find the continuous rate over that period, and work out the half life.

Example: Material X decayed from 53kg to 37kg over 9 months. What's the continuous decay rate and half life (in years)?

The ratio between new and old was 37/53, so ln(37/53) = -.359 = -35.9% continuous growth over our time period. This happened over 9 months, so the monthly continuous rate is -35.9/9 = -3.98%. Scaling this up, the yearly continuous rate is -3.98% * 12 = -47.9%. (Notice how the rate must be scaled to match the time period. Earning "12% interest" isn't helpful without a time period. "12% interest per day" is different than "12% interest per year".)

Now that we know the continuous rate is -47.9% per year, we can work out how long until we're at 50%:


\begin{aligned}
e^{{-.479} \cdot \text{years}} &= 0.5 \\
\ln(e^{{-.479} \cdot \text{years}}) &= \ln(0.5) \\
{-.479} \cdot \text{years} &= {-.693} \\
\text{years} &= {-.693} / {-.479} = 1.44
\end{aligned}

The half-life is 1.44 years.

Example: Stock Market Growth

This is a tricky one: the stock market changes every day, so it seems like it'd continuous, but there isn't an underlying predictable rate. We see a lot of jumpy changes, and sample them at yearly intervals to see how we're doing. The market is usually described with an annual average growth rate:

\displaystyle{
\text{expected value} = \text{investment} \cdot (1 + 8\%)^{\text{years}}
}

A continuous rate of the form $e^x$ doesn't really make sense for the system. We aren't trying to model our portfolio's value on a per-instant basis: we want to know what to expect in 30 years.

Example: Population Growth

Population is tricky: depending on the animal, discrete or continuous model can make sense.

A bacteria colony is made of billions of organisms. Although each bacteria cell grows discretely (it has to wait until it splits before splitting again), the entire colony grows smoothly because so many bacteria are in different stages of growth.

Like the radioactive decay example, we can sample the colony at different time periods and work out how long it takes to double. We might have a continuous rate ($e^x$) that expresses the colony's instant rate, and a discrete rate ($2^x$) that helps us humans count the doublings.

One of my pet peeves were problems like "A bacteria colony doubles after 24 hours...". Argh! Are you telling me the bacteria colony just happens to have a continuous rate of precisely ln(2) over the course of a day?

I'd prefer you told me the colony doubled while a grad student stared at a petri dish for 24 hours straight. (1.98kg... 1.99kg... 2.00kg. I found the doubling time, I can go home! What's that Professor? I...ok, I'll work out the continuous rate after an hour next time.)

Rant aside, how about modeling a tiger population? Tigers have breeding seasons. They aren't having kids throughout the year, so the population changes in a discrete event.

\displaystyle{
\text{new population} = \text{current population} * (1 + \text{growth rate})^{\text{years}}
}

(The model gets more complex as you account for how long it takes for cubs to have children of their own.)

Onward and Upward

I wrote this post because my video on e had questions about how $2^x$ represents "staircase growth". Isn't that a smooth curve too?

Sure, but most of the time we use 2 as a base to model discrete patterns. $2^n$ (where n is an integer) models discrete scenarios like coin flips or binary digits. If your system does change continuously, why not provide the continuous rate and write $e^{\ln(2) x}$?

There's no right and wrong here, just the message we convey. A whole-number base ($2^x$, $3^x$) implies you want people to think about whole-number values of x (and half-life is a good example). Using $e$ as a base ($e^{\text{rate} \cdot \text{time}}$) implies you want people to think about change that happens at every moment.

Either way, be fluent in both models and learn to hop between the two.

Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

How To Think With Exponents And Logarithms

Here’s a trick for thinking through problems involving exponents and logs. Just ask two questions:

1) Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)

  • Logarithms reveal the inputs that caused the growth
  • Exponents find the final result of growth

2) Are we talking about the grower’s perspective, or an observer’s?

  • e and the natural log are from the grower’s instant-by-instant perspective
  • Base 10, Base 2, etc. are measurements convenient for a human observer

In my head, I put the options in a table:

exponent logarithm point of view comparison

I have thoughts like “I need the cause, from the grower’s perspective… that’s the natural log.”. (Natural log is abbreviated with lowercase LN, from the high-falutin’ logarithmus naturalis.)

I was frustrated with classes that described the inner part of the table, the raw functions, without the captions that explained when to use them!

That won’t fly, let’s get direct practice thinking with logs and exponents.

Scenario: Describing GDP Growth

Here’s a typical example of growth:

  • From 2000 to 2010, the US GDP changed from 9.9 trillion to 14.4 trillion

Ok, sure, those numbers show change happened. But we probably want insight into the cause: What average annual growth rate would account for this change?

Immediately, my brain thinks “logarithms” because we’re working backwards from the growth to the rate that caused it. I start with a thought like this:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

A good start, but let’s sharpen it up.

First, which logarithm should we use?

By default, I pick the natural logarithm. Most events end up being in terms of the grower (not observer), and I like “riding along” with the growing element to visualize what’s happening. (Radians are similar: they measure angles in terms of the mover.)

Next question: what change do we apply the logarithm to?

We’re really just interested in the ratio between start and finish: 9.9 trillion to 14.4 trillion in 10 years. This is the same growth rate as going from \$9.90 to \$14.40 in the same period.

We can sharpen our thought:

\displaystyle{\text{natural logarithm of growth ratio} \rightarrow \text{cause of growth} }

\displaystyle{\ln(\frac{14.4}{9.9}) = .374}

Ok, the cause was a rate of .374 or 37.4%. Are we done?

Not yet. Logarithms don’t know about how long a change took (we didn’t plug in 10 years, right?). They give us a rate as if all the change happened in a single time period.

The change could indeed be a single year of 37.4% continuous growth, or 2 years of 18.7% growth, or some other combination.

From the scenario, we know the change took 10 years, so the rate must have been:

\displaystyle{ \text{rate} = \frac{.374}{10} = .0374 = 3.74\%}

From the viewpoint of instant, continuous growth, the US economy grew by 3.74% per year.

Are we done now? Not quite!

This continuous rate is from the grower’s perspective, as if we’re “riding along” with the economy as it changes. A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:

\displaystyle{\text{exponent with rate and time} \rightarrow \text{effect of growth} }

\displaystyle{e^{\text{rate} \cdot \text{time}} = \text{growth}}

\displaystyle{e^{.0374 \cdot 1} = 1.0381}

The year-over-year gain is 3.8%, slightly higher than the 3.74% instantaneous rate due to compounding. Here’s another way to put it:

  • From an instant-by-instant basis, a given part of the economy is growing by 3.74%, modeled by $e^(.0374 * years)$
  • On a year-by-year basis, with compounding effects worked out, the economy grows by 3.81%, modeled by $1.0381^years$

In finance, we may want the year-over-year change which can be compared nicely with other trends. In science and engineering, we prefer modeling behavior on an instantaneous basis.

Scenario: Describing Natural Growth

I detest contrived examples like “Assume bacteria doubles every 24 hours, find its growth formula.”. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling?

A better scenario: “Hey, I found some bacteria, waited an hour, and the lump grew from 2.3 grams to 2.32 grams. I’m going to lunch now. Figure out how much we’ll have when I’m back in 3 hours.”

Let’s model this. We’ll need a logarithm to find the growth rate, and then an exponent to project that growth forward. Like before, let’s keep everything in terms of the natural log to start.

The growth factor is:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

\displaystyle{\ln(\text{growth}) = \ln(2.32/2.3) = .0086 = .86\%}

That’s the rate for one hour, and the general model to project forward will be

\displaystyle{\text{exponent with rate and time} \rightarrow \text{effect of growth} }

\displaystyle{e^{.0086 \cdot \text{hours}} \rightarrow \text{effect of growth} }

If we start with 2.32 and grow for 3 hours we’ll have:

\displaystyle{2.32 \cdot e^{.0086 \cdot 3} = 2.38}

Just for fun, how long until the bacteria doubles? Imagine waiting for 1 to turn to 2:

\displaystyle{1 \cdot e^{.0086 \cdot \text{hours}} = 2}

We can mechanically take the natural log of both sides to “undo the exponent”, but let’s think intuitively.

If 2 is the final result, then ln(2) is the growth input that got us there (some rate × time). We know the rate was .0086, so the time to get to 2 would be:

\displaystyle{ \text{hours} = \frac{\ln(2)}{\text{rate}} = \frac{.693}{.0086} = 80.58}

The colony will double after ~80 hours. (Glad you didn’t stick around?)

What Does The Perspective Change Really Mean?

Figuring out whether you want the input (cause of growth) or output (result of growth) is pretty straightforward. But how do you visualize the grower’s perspective?

Imagine we have little workers who are building the final growth pattern (see the article on exponents):

compound interest

If our growth rate is 100%, we’re telling our initial worker (Mr. Blue) to work steadily and create a 100% copy of himself by the end of the year. If we follow him day-by-day, we see he does finish a 100% copy of himself (Mr. Green) at the end of the year.

But… that worker he was building (Mr. Green) starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work (same annual rate as Mr. Blue) and he builds Mr. Red. Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months.

What if Mr. Green showed up after 4 months? A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by $e^x$:

continuous growth

The natural log gives a growth rate in terms of an individual worker’s perspective. We plug that rate into $e^x$ to find the final result, with all compounding included.

Using Other Bases

Switching to another type of logarithm (base 10, base 2, etc.) means we’re looking for some pattern in the overall growth, not what the individual worker is doing.

Each logarithm asks a question when seeing a change:

  • Log base e: What was the instantaneous rate followed by each worker?
  • Log base 2: How many doublings were required?
  • Log base 10: How many 10x-ings were required?

Here’s a scenario to analyze:

  • Over 30 years, the transistor counts on typical chips went from 1000 to 1 billion

How would you analyze this?

  • Microchips aren’t a single entity that grow smoothly over time. They’re separate editions, from competing companies, and indicate a general tech trend.
  • Since we’re not “riding along” with an expanding microchip, let’s use a scale made for human convenience. Doubling is easier to think about than 10x-ing.

With these assumptions we get:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

\displaystyle{\log_2(\frac{\text{1 billion}}{1000}) = \log_2(\text{1 million}) \sim \text{20 doublings}}

The “cause of growth” was 20 doublings, which we know occurred over 30 years. This averages 2/3 doublings per year, or 1.5 years per doubling — a nice rule of thumb.

From the grower’s perspective, we’d compute $\ln(\text{1 billion}/1000) / \text{30 years} = 46\%$ continuous growth (a bit harder to relate to in this scenario).

We can summarize our analysis in a table:

exponents transistor example

Summary

Learning is about finding the hidden captions behind a concept. When is it used? What point view does it bring to the problem?

My current interpretation is that exponents ask about cause vs. effect and grower vs. observer. But we’re never done; part of the fun is seeing how we can recaption old concepts.

Happy math.

Appendix: The Change Of Base Formula

Here’s how to think about switching bases. Assuming a 100% continuous growth rate,

  • ln(x) is the time to grow to x
  • ln(2) is the time to grow to 2

Since we have the time to double, we can see how many would “fit” in the total time to grow to x:

\displaystyle{\text{number of doublings from 1 to x} = \frac{\ln(x)}{\ln(2)} = \log_2(x)}

For example, how many doublings occur from 1 to 64?

Well, ln(64) = 4.158. And ln(2) = .693. The number of doublings that fit is:

\displaystyle{\frac{\ln(64)}{\ln(2)} = \frac{4.158}{.693} = 6}

In the real world, calculators may lose precision, so use a direct log base 2 function if possible. And of course, we can have a fractional number: Getting from 1 to the square root of 2 is “half” a doubling, or log2(1.414) = 0.5.

Changing to log base 10 means we’re counting the number of 10x-ings that fit:

\displaystyle{\text{number of 10x-ings from 1 to x} = \frac{\ln(x)}{\ln(10)} = \log_{10}(x) }

Neat, right? Read Using Logarithms in the Real World for more examples.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Using Logarithms in the Real World

Logarithms are everywhere. Ever use the following phrases?

  • 6 figures
  • Double digits
  • Order of magnitude
  • Interest rate

You're describing numbers in terms of their powers of 10, a logarithm. And an interest rate is the logarithm of the growth in an investment.

Surprised that logarithms are so common? Me too. Most attempts at Math In the Real World (TM) point out logarithms in some arcane formula, or pretend we're geologists fascinated by the Richter Scale. "Scientists care about logs, and you should too. Also, can you imagine a world without zinc?"

No, no, no, no no, no no! (Mama mia!)

Math expresses concepts with notation like "ln" or "log". Finding "math in the real world" means encountering ideas in life and seeing how they could be written with notation. Don't look for the literal symbols! When was the last time you wrote a division sign? When was the last time you chopped up some food?

Ok, ok, we get it: what are logarithms about?

Logarithms find the cause for an effect, i.e the input for some output

A common "effect" is seeing something grow, like going from \$100 to \$150 in 5 years. How did this happen? We're not sure, but the logarithm finds a possible cause: A continuous return of ln(150/100) / 5 = 8.1% would account for that change. It might not be the actual cause (did all the growth happen in the final year?), but it's a smooth average we can compare to other changes.

By the way, the notion of "cause and effect" is nuanced. Why is 1000 bigger than 100?

  • 100 is 10 which grew by itself for 2 time periods ($10 * 10$)
  • 1000 is 10 which grew by itself for 3 time periods ($10 * 10 * 10$)

We can think of numbers as outputs (1000 is "1000 outputs") and inputs ("How many times does 10 need to grow to make those outputs?"). So,

1000 outputs > 100 outputs

because

3 inputs > 2 inputs

Or in other words:

log(1000) > log(100)

Why is this useful?

Logarithms put numbers on a human-friendly scale.

Large numbers break our brains. Millions and trillions are "really big" even though a million seconds is 12 days and a trillion seconds is 30,000 years. It's the difference between an American vacation year and the entirety of human civilization.

The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" (i.e. their power base 10). This smaller scale (0 to 100) is much easier to grasp:

  • power of 0 = $10^0$ = 1 (single item)
  • power of 1 = $10^1$ = 10
  • power of 3 = $10^3$ = thousand
  • power of 6 = $10^6$ = million
  • power of 9 = $10^9$ = billion
  • power of 12 = $10^12$ = trillion
  • power of 23 = $10^23$ = number of molecules in a dozen grams of carbon
  • power of 80 = $10^80$ = number of molecules in the universe

A 0 to 80 scale took us from a single item to the number of things in the universe. Not too shabby.

Logarithms count multiplication as steps

Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger. With the natural log, each step is "e" (2.71828...) times more.

When dealing with a series of multiplications, logarithms help "count" them, just like addition counts for us when effects are added.

Show me the math

Time for the meat: let's see where logarithms show up!

Six-figure salary or 2-digit expense

We're describing numbers in terms of their digits, i.e. how many powers of 10 they have (are they in the tens, hundreds, thousands, ten-thousands, etc.). Adding a digit means "multiplying by 10", i.e.

\displaystyle{1 \text{[1 digit]} \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \text{[5 more digits]} = 10^5 = 100,000}

Logarithms count the number of multiplications added on, so starting with 1 (a single digit) we add 5 more digits ($10^5$) and 100,000 get a 6-figure result. Talking about "6" instead of "One hundred thousand" is the essence of logarithms. It gives a rough sense of scale without jumping into details.

Bonus question: How would you describe 500,000? Saying "6 figure" is misleading because 6-figures often implies something closer to 100,000. Would "6.5 figure" work?

Not really. In our heads, 6.5 means "halfway" between 6 and 7 figures, but that's an adder's mindset. With logarithms a ".5" means halfway in terms of multiplication, i.e the square root ($9^.5$ means the square root of 9 -- 3 is halfway in terms of multiplication because it's 1 to 3 and 3 to 9).

Taking log(500,000) we get 5.7, add 1 for the extra digit, and we can say "500,000 is a 6.7 figure number". Try it out here:

Order of magnitude

We geeks love this phrase. It means roughly "10x difference" but just sounds cooler than "1 digit larger".

In computers, where everything is counted with bits (1 or 0), each bit has a doubling effect (not 10x). So going from 8 to 16 bits is "8 orders of magnitude" or $2^8 = 256$ times larger. ("Larger" in this case refers to the amount of memory that can be addressed.) Going from 16 to 32 bits means an extra 16 orders of magnitude, or $2^16$ ~ 65,536 times more memory that can be addressed.

Interest Rates

How do we figure out growth rates? A country doesn't intend to grow at 8.56% per year. You look at the GDP one year and the GDP the next, and take the logarithm to find the implicit growth rate.

My two favorite interpretations of the natural logarithm (ln(x)), i.e. the natural log of 1.5:

  • Assuming 100% growth, how long do you need to grow to get to 1.5? (.405, less than half the time period)
  • Assuming 1 unit of time, how fast do you need to grow to get to 1.5? (40.5% per year, continuously compounded)

Logarithms are how we figure out how fast we're growing.

Measurement Scale: Google PageRank

Google gives every page on the web a score (PageRank) which is a rough measure of authority / importance. This is a logarithmic scale, which in my head means "PageRank counts the number of digits in your score".

So, a site with pagerank 2 ("2 digits") is 10x more popular than a PageRank 1 site. My site is PageRank 5 and CNN has PageRank 9, so there's a difference of 4 orders of magnitude ($10^4$ = 10,000).

Roughly speaking, I get about 7000 visits / day. Using my envelope math, I can guess CNN gets about 7000 * 10,000 = 70 million visits / day. (How'd I do that? In my head, I think $7k * 10k = 70 * k * k = 70 * M$). They might have a few times more than that (100M, 200M) but probably not up to 700M.

Google conveys a lot of information with a very rough scale (1-10).

Measurement Scale: Richter, Decibel, etc.

Sigh. We're at the typical "logarithms in the real world" example: Richter scale and Decibel. The idea is to put events which can vary drastically (earthquakes) on a single scale with a small range (typically 1 to 10). Just like PageRank, each 1-point increase is a 10x improvement in power. The largest human-recorded earthquake was 9.5; the Yucatán Peninsula impact, which likely made the dinosaurs extinct, was 13.

Decibels are similar, though it can be negative. Sounds can go from intensely quiet (pindrop) to extremely loud (airplane) and our brains can process it all. In reality, the sound of an airplane's engine is millions (billions, trillions) of times more powerful than a pindrop, and it's inconvenient to have a scale that goes from 1 to a gazillion. Logs keep everything on a reasonable scale.

Logarithmic Graphs

You'll often see items plotted on a "log scale". In my head, this means one side is counting "number of digits" or "number of multiplications", not the value itself. Again, this helps show wildly varying events on a single scale (going from 1 to 10, not 1 to billions).

Moore's law is a great example: we double the number of transistors every 18 months (image courtesy Wikipedia).

Moore's Law

The neat thing about log-scale graphs is exponential changes (processor speed) appear as a straight line. Growing 10x per year means you're steadily marching up the "digits" scale.

Onward and upward

If a concept is well-known but not well-loved, it means we need to build our intuition. Find the analogies that work, and don't settle for the slop a textbook will trot out. In my head:

  • Logarithms find the root cause for an effect (see growth, find interest rate)
  • They help count multiplications or digits, with the bonus of partial counts (500k is a 6.7 digit number)

Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Understanding Exponents (Why does 0^0 = 1?)

We’re taught that exponents are repeated multiplication. This is a good introduction, but it breaks down on 3^1.5 and the brain-twisting 0^0. How do you repeat zero zero times and get 1?

You can’t, not while exponents are repeated multiplication. Today our mental model is due for an upgrade.

addition multiplication mental model

Viewing arithmetic as transformations

Let’s step back — how do we learn arithmetic? We’re taught that numbers are counts of something (fingers), addition is combining counts (3 + 4 = 7) and multiplication is repeated addition (2 times 3 = 2 + 2 + 2 = 6).

Repeated addition works when multiplying by nice round numbers like 2 and 10, but not when using numbers like -1 and $\sqrt{2}$. Why?

Our model was incomplete. Numbers aren’t just a count; a better viewpoint is a position on a line. This position can be negative (-1), between other numbers ($\sqrt{2}$), or in another dimension (i).

Arithmetic became a way to transform a number: Addition was sliding (+3 means slide 3 units to the right), and multiplication was scaling (times 3 means scale it up 3x).

So what are exponents?

Enter the Expand-o-tron(TM)

Let me introduce the Expand-o-tron 3000.

expand o tron example

Yes, this device looks like a shoddy microwave — but instead of heating food, it grows numbers. Put a number in and a new one comes out. Here’s how:

  • Start with 1.0
  • Set the growth to the desired change after one second (2x, 3x, 10.3x)
  • Set the time to the number of seconds
  • Push the button

And shazam! The bell rings and we pull out our shiny new number. Suppose we want to change 1.0 into 9:

  • Put 1.0 in the expand-o-tron
  • Set the change for “3x” growth, and the time for 2 seconds
  • Push the button

The number starts transforming as soon as we begin: We see 1.0, 1.1, 1.2… and just as finish the first second, we’re at 3.0. But it keeps going: 3.1, 3.5, 4.0, 6.0, 7.5. As just as we finish the 2nd second we’re at 9.0. Behold our shiny new number!

Mathematically, the expand-o-tron (exponent function) does this:

\displaystyle{\text{original} \cdot \text{growth}^{\text{duration}} = \text{new}}

or

\displaystyle{\text{growth}^{\text{duration}} = \frac{\text{new}}{\text{original}}}

For example, 3^2 = 9/1. The base is the amount to grow each unit (3x), and the exponent is the amount of time (2). A formula like 2^n means “Use the expand-o-tron at 2x growth for n seconds”.

We always start with 1.0 in the expand-o-tron to see how it changes a single unit. If we want to see what would happen if we started with 3.0 in the expand-o-tron, we just scale up the final result. For example:

  • “Start with 1 and double 3 times” means 1 * 2^3 = 1 * 2 * 2 * 2 = 8
  • “Start with 3 and double 3 times” means 3 * 2^3 = 3 * 2 * 2 * 2 = 24

Whenever you see an plain exponent by itself (like 2^3), we’re starting with 1.0.

Understanding the Exponential Scaling Factor

When multiplying, we can just state the final scaling factor. Want it 8 times larger? Multiply by 8. Done.

Exponents are a bit… finicky:

You: I’d like to grow this number.

Expand-o-tron: Ok, stick it in.

You: How big will it get?

Expand-o-tron: Gee, I dunno. Let’s find out…

You: Find out? I was hoping you’d kn-

Expand-o-tron: Shh!!! It’s growing! It’s growing!

You:

Expand-o-tron: It’s done! My masterpiece is alive!

You: Can I go now?

The expand-o-tron is indirect. Just looking at it, you’re not sure what it’ll do: What does 3^10 mean to you? How does it make you feel? Instead of a nice tidy scaling factor, exponents want us to feel, relive, even smell the growing process. Whatever you end with is your scaling factor.

It sounds roundabout and annoying. You know why? Most things in nature don’t know where they’ll end up!

Do you think bacteria plans on doubling every 14 hours? No — it just eats the moldy bread you forgot about in the fridge as fast as it can, and as it gets more it starts growing even faster. To predict the behavior, we use how fast they’re growing (current rate) and how long they’ll be changing (time) to figure out their final value.

The answer has to be worked out — exponents are a way of saying “Begin with these conditions, start changing, and see where you end up”. The expand-o-tron (or our calculator) does the work by crunching the numbers to get the final scaling factor. But someone has to do it.

Understanding Fractional Powers

Let’s see if the expand-o-tron can help us understand exponents. First up: what does at 2^1.5 mean?

It’s confusing when we think of repeated multiplication. But the expand-o-tron makes it simple: 1.5 is just the amount of time in the machine.

  • 2^1 means 1 second in the machine (2x growth)
  • 2^2 means 2 seconds in the machine (4x growth)

2^1.5 means 1.5 seconds in the machine, so somewhere between 2x and 4x growth (more later). The idea of “repeated counting” had us stuck using whole numbers, but fractional seconds are completely fine.

Multiplying exponents

What if we want to two growth cycles back-to-back? Let’s say we use the machine for 2 seconds, and then use it for 3 seconds at the exact same power:

\displaystyle{x^2 \cdot x^3 = ?}

Think about your regular microwave — isn’t this the same as one continuous cycle of 5 seconds? It sure is. As long as the power setting (base) stayed the same, we can just add the time:

\displaystyle{x^y \cdot x^z = x^{y + z}}

Again, the expand-o-tron gives us a scaling factor to change our number. To get the total effect from two consecutive uses, we just multiply the scaling factors together.

Square roots

Let’s keep going. Let’s say we’re at power level a and grow for 3 seconds:

\displaystyle{a^3}

Not too bad. Now what would growing for half that time look like? It’d be 1.5 seconds:

\displaystyle{a^{1.5}}

Now what would happen if we did that twice?

\displaystyle{a^{1.5} \cdot a^{1.5} = a^3}

\displaystyle{\text{partial growth} \cdot \text{partial growth} = \text{full growth}}

Looking at this equation, we see “partial growth” is the square root of full growth! If we divide the time in half we get the square root scaling factor. And if we divide the time in thirds?

\displaystyle{a^1 \cdot a^1 \cdot a^1 = a^3}

\displaystyle{\text{partial growth} \cdot \text{partial growth} \cdot \text{partial growth} = \text{full growth}}

And we get the cube root! For me, this is an intuitive reason why dividing the exponents gives roots: we split the time into equal amounts, so each “partial growth” period must have the same effect. If three identical effects are multiplied together, it means they’re each a cube root.

Negative exponents

Now we’re on a roll — what does a negative exponent mean? Negative seconds means going back in time! If going forward grows by a scaling factor, going backwards should shrink by it.

\displaystyle{2^{-1} = \frac{1}{2^1}}

The sentence means “1 second ago, we were at half our current amount (1/2^1)”. In fact, this is a neat part of any exponential graph, like 2^x:

modeling 2 to the x

Pick a point like 3.5 seconds (2^3.5 = 11.3). One second in the future we’ll be at double our current amount (2^4.5 = 22.5). One second ago we were at half our amount (2^2.5 = 5.65).

This works for any number! Wherever 1 million is, we were at 500,000 one second before it. Try it below:

Taking the zeroth power

Now let’s try the tricky stuff: what does 3^0 mean? Well, we set the machine for 3x growth, and use it for… zero seconds. Zero seconds means we don’t even use the machine!

Our new and old values are the same (new = old), so the scaling factor is 1. Using 0 as the time (power) means there’s no change at all. The scaling factor is always 1.

Taking zero as a base

How do we interpret 0^x? Well, our growth amount is “0x” — after a second, the expand-o-tron obliterates the number and turns it to zero. But if we’ve obliterated the number after 1 second, it really means any amount of time will destroy the number:

0^(1/n) = nth root of 0^1 = nth root of 0 = 0

No matter the tiny power we raise it to, it will be some root of 0.

Zero to the zeroth power

At last, the dreaded 0^0. What does it mean?

The expand-o-tron to the rescue: 0^0 means a 0x growth for 0 seconds!

Although we planned on obliterating the number, we never used the machine. No usage means new = old, and the scaling factor is 1. 0^0 = 1 * 0^0 = 1 * 1 = 1 — it doesn’t change our original number. Mystery solved!

(For the math geeks: Defining 0^0 as 1 makes many theorems work smoothly. In reality, 0^0 depends on the scenario (continuous or discrete) and is under debate. The microwave analogy isn’t about rigor — it helps me see why it could be 1, in a way that “repeated counting” does not.)

Here's what happens when we try to plug in actual numbers:

Advanced: Repeated Exponents (a to the b to the c)

Repeated exponents are tricky. What does

\displaystyle{\Large (2^a)^b}

mean? It’s “repeated multiplication, repeated” — another way of saying “do that exponent thing once, and do it again”. Let’s dissect it:

\displaystyle{(2^3)^4}

  • First, I want to grow by doubling each second: do that for 3 seconds (2^3)
  • Then, whatever my number is (8x), I want to grow by that new amount for 4 seconds (8^4)

The first exponent (^3) just knows to take “2″ and grow it by itself 3 times. The next exponent (^4) just knows to take the previous amount (8) and grow it by itself 4 times. Each time unit in “Phase II” is the same as repeating all of Phase I:

\displaystyle{(2^3)^4 = 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3 = 2^{3+3+3+3} = 2^{12} }

This is where the repeated counting interpretation helps get our bearings. But then we bring out the expand-o-tron: we grow for 3 seconds in Phase I, and redo that for 4 more seconds. It works for fractional powers — for example,

\displaystyle{(2^{3.1})^{4.2}}

means “Grow for 3.1 seconds, and use that new growth rate for 4.2 seconds”. We can smush together the time (3.1 × 4.2) like this:

\displaystyle{(a^b)^c = a^{b\cdot c} = (a^c)^b}

It’s different, so try some examples:

  • (2^1)^x means “Grow at 2 for 1 second, and ‘do that growth’ for x more seconds”.
  • 7 = (7^0.5)^2 means “We can jump to 7 all at once. Or, we can plan on growing to 7 but only use half the time ($\sqrt{7}$). But we can do that process for 2 seconds, which gives us the full amount ($\sqrt{7}^2 = 7$).”

We’re like kids learning that 3 times 7 = 7 times 3. (Or that a% of b = b% of a — it’s true!).

Advanced: Rewriting Exponents For The Grower

The expand-o-tron is a bit strange: numbers start growing the instant they’re inside, but we specify the desired growth at the end of each second.

We say we want 2x growth at the end of the first second. But how do we know what rate to start off with? How fast should we be growing at 0.5 seconds? It can’t be the full amount, or else we’ll overshoot our goal as our interest compounds.

Here’s the key: Growth curves written like 2^x are from the observer’s viewpoint, not the grower.

The value “2″ is measured at the end of the interval and we work backwards to create the exponent. This is convenient for us, but not the growing quantity — bacteria, radioactive elements and money don’t care about lining up with our ending intervals!

No, these critters know their current, instantaneous growth rate, and don’t try to line it up with our boundaries. It’s just like understanding radians vs. degrees — radians are “natural” because they are measured from the mover’s viewpoint.

To get into the grower’s viewpoint, we use the magical number e. There’s much more to say, but we can convert any “observer-focused” formula like 2^x into a “grower-focused” one:

\displaystyle{2^x = (e^{\ln(2)})^x = e^{\ln(2)x} }

In this case, ln(2) = .693 = 69.3% is the instantaneous growth rate needed to look like 2^x to an observer. When you enter “2x growth at the end of each period”, the expand-o-tron knows to grow the number at a rate of 69.3%.

We’ll save these details for another day — just remember the difference between the grower’s instantaneous growth rate (which the bacteria controls) and the observer’s chart that’s measured at the end of each interval. Underneath it all, every exponential curve is a scaled version of e^x:

\displaystyle{a^x = (e^{\ln(a)})^x = e^{\ln(a)x} }

Every exponent is a variation of e, just like every number is a scaled version of 1.

Why use this analogy?

Does the expand-o-tron exist? Do numbers really gather up in a line? Nope — they’re ways of looking at the world.

The expand-o-tron removes the mental hiccups when seeing 2^1.5 or even 0^0: it’s just 0x growth for 0 seconds, which doesn’t change the number. Everything from slide rules to Euler’s formula begins to click once we recognize the core theme of growth — even beasts like i^i can be tamed.

Friends don’t let friends think of exponents as repeated multiplication. Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

A Visual Guide to Simple, Compound and Continuous Interest Rates

Interest rates are confusing, despite their ubiquity. This post takes an in-depth look at why interest rates behave as they do.

Understanding these concepts will help understand finance (mortgages & savings rates), along with the omnipresent e and natural logarithm. Here’s our cheatsheet:

Term Formula Description & Usage
Simple \displaystyle{P \cdot (1 + r \cdot n)} Fixed, non-growing return (bond coupons)
Compound
(Annual)
\displaystyle{P \cdot (1 + r)^n} Changes each year (stock market, inflation)
Compound
(n times per year)
\displaystyle{P \cdot (1 + r/n)^{nt}} Changes each month/week/day (savings account)
Continuous Growth \displaystyle{P \cdot e^{rt}} Changes each instant (radioactive decay, temperature)
APR Annual Percentage Rate (compounding not included)
APY Annual Percentage Yield (all compounding effects included)
  • P = principal, your initial investment (i.e., \$1,000)
  • r = interest rate (i.e., 5% per year)
  • n = number of time periods (i.e., 3 years)

And a quick calculator to convert APR to APY:

Why the fuss?

Interest rates are complex. Like Roman numerals and hieroglyphics, our first system “worked” but wasn’t quite ideal.

In the beginning, you might have had 100 gold coins and were paid 12% per year (percent = per cent = per hundred — those Roman numerals still show up!). It’s simple enough: we get 12 coins a year. But is it really 12?

If we break it down, it seems we earn 1 gold a month: 6 for January-June, and 6 for July-December. But wait a minute — after our June payout we’d have 106 gold in July, and yet earn only 6 during the rest of the year? Are you saying 100 and 106 earn the same amount in 6 months? By that logic, do 100 and 200 earn the same amount, too? Uh oh.

This issue didn’t seem to bother the ancient Egyptians, but did raise questions in the 1600s and led to Bernoulli’s discovery of e (sorry math fans, e wasn’t discovered via some hunch that a strange limit would have useful properties). There’s much to say about this riddle — just keep this in mind as we dissect interest rates:

  • Interest rates and terminology were invented before the idea of compounding. Heck, loans were around in 1500 BC, before exponents, 0, or even the decimal point! So it’s no wonder our discussions can get confusing.
  • Nature doesn’t wait for a human year before changing. Interest earnings are a type of “growth”, but natural phenomena like temperature and radioactive decay change constantly, every second and faster. This is one reason why physics equations model change with “e” and not “$(1+r)^n$”: Nature rudely ignores our calendar when making adjustments.

Learn the Lingo

As a result of these complications, we need a few terms to discuss interest rates:

  • APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula.
  • APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken into account. You can consider this “total return” in the formula. The APY is greater than or equal to the APR.

APR is what the bank tells you, the APY is what you pay (the price after taxes, shipping and handling, if you get my drift). And of course, banks advertise the rate that looks better.

Getting a credit card or car loan? They’ll show the “low APR” you’re paying, to hide the higher APY. But opening a savings account? Well, of course they’d tout the “high APY” they’re paying to look generous.

The APY (actual yield) is what you care about, and the way to compare competing offers.

Simple Interest

Let’s start on the ground floor: Simple interest pays a fixed amount over time. A few examples:

  • Aesop’s fable of the golden goose: every day it laid a single golden egg. It couldn’t lay faster, and the eggs didn’t grow into golden geese of their own.
  • Corporate bonds: A bond with a face value of \$1000 and 5% interest rate (coupon) pays you \$50 per year until it expires. You can’t increase the face value, so \$50/year is what you will get from the bond. (In reality, the bond would pay \$25 every 6 months).

Simple interest is the most basic type of return. Depositing \$100 into an account with 50% simple (annual) interest looks like this:

simple interest

You start with a principal (aka investment) of \$100 and earn \$50 each year. I imagine the blue principal “shoveling” green money upwards every year.

However, this new, green money is stagnant — it can’t grow! With simple interest, the \$50 just sits there. Only the original \$100 can do “work” to generate money.

Simple interest has a simple formula: Every period you earn P * r (principal * interest rate). After n periods you have:

\displaystyle{\text{balance} = P \cdot r \cdot n}

This formula works as long as “r” and “n” refer to the same time period. It could be years, months, or days — though in most cases, we’re considering annual interest. There’s no trickery because there’s no compounding — interest can’t grow.

Simple interest is useful when:

  • Your interest earnings create something that cannot grow more. It’s like the golden goose creating eggs, or a corporate bond paying money that cannot be reinvested.
  • You want simple, predictable, non-exponential results. Suppose you’re encouraging your kids to save. You could explain that you’ll put aside \$1/month in “fun money” for every \$20 in their piggybank. Most kids would be thrilled and buy comic books each month. If your last name is Greenspan, your kid might ask to reinvest the dividend.

In practice, simple interest is fairly rare because most types of earnings can be reinvested. There really isn’t an APR vs APY distinction, since your earnings can’t change: you always earn the same amount per year.

Really Understanding Growth

Most interest explanations stop there: here’s the formula, now get on your merry way. Not here: let’s see what’s really happening.

First, what does an interest rate mean? I think of it as a type of “speed”:

  • 50 mph means you’ll travel 50 miles in the course of an hour
  • r = 50% per year means you’ll earn 50% of your principal in the course of a year. If P = \$100, you’ll earn \$50/year (your “speed of money growth”).

But both types of speed have a subtlety: we don’t have to wait the full time period!

Does driving 50 mph mean you must go a full hour? No way! You can drive “only” 30 minutes and go 25 miles (50 mph * .5 hours). You could drive 15 minutes and go 12.5 miles (50 mph * .25 hours). You get the idea.

Interest rates are similar. An interest rate gives you a “trajectory” or “pace” to follow. If you have \$100 at a 50% simple interest rate, your pace is \$50/year. But you don’t need to follow that pace for a full year! If you grew for 6 months, you should be entitled to \$25. Take a look at this:

simple interest trajectory

We start with \$100, in blue. Each year that blue contributes \$50 (in green) to our total amount. Of course, with simple interest our earnings are based on our original amount, not the “new total”. Connecting the dots gives us a trendline: we’re following a path of \$50/year. Our payouts look like a staircase because we’re only paid at the end of the year, but the trajectory still works.

Simple interest keeps the same trajectory: we earn “P*r” each year, no matter what (\$50/year in this case). That straight line perfectly predicts where we’ll end up.

The idea of “following a trajectory” may seem strange, but stick with it — it will really help when understanding the nature of e.

One point: the trajectory is “how fast” a bank account is growing at a certain moment. With simple interest, we’re stuck in a car going the same speed: \$50/year, or 50 mph. In other cases, our rate may change, like a skydiver: they start off slow, but each second fall faster and faster. But at any instant, there’s a single speed, a single trajectory.

(The math gurus will call this trajectory a “derivative” or “gradient”. No need to hit a mosquito with the calculus sledgehammer just yet.)

Basic Compound Interest

Simple interest should make you squirm. Why can’t our interest earn money? We should use the bond payouts (\$50/year) to buy more bonds. Heck, we should use the golden eggs to fund research into cloning golden geese!

Compound growth means your interest earns interest. Einstein called it “one of the most powerful forces in nature”, and it’s true. When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast.

The most basic type is period-over-period return, which usually means “year over year”. Reinvesting our interest annually looks like this:

compound interest graph

We earn \$50 from year 0 – 1, just like with simple interest. But in year 1-2, now that our total is \$150, we can earn \$75 this year (50% * 150) giving us \$225. In year 2-3 we have \$225, so we earn 50% of that, or \$112.50.

In general, we have (1 + r) times more “stuff” each year. After n years, this becomes:

\displaystyle{\text{balance} = P \cdot (1 + r)^n}

Exponential growth outpaces simple, linear interest, which only had \$250 in year 3 (100 + 3*50). Compound growth is useful when:

  • Interest can be reinvested, which is the case for most savings accounts.
  • You want to predict a future value based on a growth trend. Most trends, like inflation, GDP growth, etc. are assumed to be “compoundable”. Yearly GDP growth of 3% over 10 years is really $(1.03)^10 = 1.344$, or a 34.4% increase over that decade.

Interest as a Factory

The typical interpretation sees money as a “blob” that grows over time. This view works, but sometimes I like to see interest earnings as a “factory” that generates more money:

compound interest factory analogy

Here’s what’s happening:

  • Year 0: We start with \$100.
  • Year 1: Our \$100 creates a \$50 “bond”.
  • Year 2: The \$100 generates another \$50 bond. The \$50 generates a \$25 bond. The total is 50 + 25 = 75, which matches up.
  • Year 3: Things get a bit crazy. The \$100 creates a third \$50 bond. The two existing \$50 bonds make \$25 each. And the \$25 makes a 12.50.
  • Years 4 to infinity: Left as an exercise for the reader. (Don’t you love that textbook cop out?)

This is an interesting viewpoint. The \$100 just mindlessly cranks out \$50 “factories”, which start earning money independently (notice the 3 blue arrows from the blue principal to the green \$50s). These \$50 factories create \$25 factories, and so on.

The pattern seems complex, but it’s simpler in a way as well. The \$100 has no idea what those zany \$50s are up to: as far as the \$100 knows, we’re only making \$50/year.

So why’s this viewpoint useful?

  • You can separate the impact of the parent (\$100) from the children. For example, at Year 3 we have \$337.50 total. The parent has earned \$150 (“3 * 50% * \$100 = \$150”, using the simple interest formula!). This means the various “children” have contributed \$337.50 – \$150 – \$100 = \$87.50, or about 1/3 the total value.
  • Breaking earnings into components helps understand e. Knowing more about e is a good thing because it shows up everywhere.

And besides, seeing old ideas in a new light is always fun. For one of us, at least.

Understanding the Trajectory

Oh, we’re not done yet. One more insight — take a look at our trajectory:

compound interest trajectory

With simple interest, we kept the same pace forever (\$50/year — pretty boring). With annually compounded interest, we get a new trajectory each year.

We deposit our money, go to sleep, and wake up at the end of the year:

  • Year 1: “Hey, waittaminute. I’ve got \$150 bucks! I should be making \$75/year, not \$50!”. You yell at your banker, crank up the dial to \$75/year, and go to sleep again.
  • Year 2: “Hey! I’ve got \$225, and should be making \$112.50 per year!”. You scream at your bank and get the rate adjusted.

This process repeats forever — we seem to never learn.

Compound Interest Revisited

Why are we waiting so long? Sure, waiting a year at a time is better than waiting “forever” (like simple interest), but I think we can do better. Let’s zoom in on a year:

simple interest gap in earnings

Look at what’s happening. The green line represents our starting pace (\$50/year), and the solid area shows the cash in our account. After 6 months, we’ve earned \$25 but don’t see a dime! More importantly, after 6 months we have the same trajectory as when we started. The interest gap shows where we’ve earned interest, but stay on our original trajectory (based on the original principal). We’re losing out on what we should be making.

Imagine I took your money and returned it after 6 months. “Well, ya see, I didn’t use it for a full year, so I don’t really owe you any interest. After all, interest is measured per year. Per yeeeeeaaaaar. Not per 6 months.” You’d smile and send Bubba to break my legs.

Annual payouts are man-made artifacts, used to keep things simple. But in reality, money should be earned all the time. We can pay interest after 6 months to reduce the gap:

compound interest twice

Here’s what happened:

  • We start with \$100 and a trajectory of \$50/year, like normal
  • After 6 months we get \$25, giving us \$125
  • We head out using the new trajectory: 50% * \$125 = \$62.5/year
  • After 6 months we collect 62.5/year times .5 year = 31.25. We have 125 + 31.25 = 156.25.

The key point is that our trajectory improved halfway through, and we earned 156.25, instead of the “expected” 150. Also, early payout gave us a smaller gap area (in white), since our \$25 of interest was doing work for the second half (it contributed the extra 6.25, or \$25 * 50% * .5 years).

For 1 year, the impact of rate r compounded n times is:

\displaystyle{(1 + r/n)^n}

In our case, we had $(1 + 50\%/2)^2$. Repeating this for t years (multiplying t times) gives:

\displaystyle{\text{balance} = P \cdot (1 + r/n)^{nt}}

Compound interest reduces the “dead space” where our interest isn’t earning interest. The more frequently we compound, the smaller the gap between earning interest and updating the trajectory.

Continuous Growth

Clearly we want money to “come online” as fast as possible. Continuous growth is compound interest on steroids: you shrink the gap into oblivion, by dividing the year into more and more time periods:

continuous growth e

The net effect is to make use of interest as soon as it’s created. We wait a millisecond, find our new sum, and go off in the new trajectory. Except it’s not every millisecond: it’s every nanosecond, picosecond, femtosecond, and intervals I don’t know the name for. Continuous growth keeps the trajectory perfectly in sync with your current amount.

Read the article on e for more details (e is a special number, like pi, and is roughly 2.718). If we have rate r and time t (in years), the result is:

\displaystyle{\text{balance} = P \cdot e^{rt}}

If you have a 50% APR, it would be an APY of $e^(.50)$ = 64.9% if compounded continuously. That’s a pretty big difference! Notice that e takes care of the icky parts, like dividing by an infinite number of periods.

Why’s this useful?

  • Most natural phenomena grow continuously. As mentioned earlier, physical phenomena grows on its own schedule: radioactive material doesn’t wait for the Earth to go around the Sun before deciding to decay. Any physical equation that models change is going to use $e^rt$.
  • $e^rt$ is the adjustable, one-size-fits-all exponential. It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest. We’ll get into this in a later article.

Most interest discussions leave e out, as continuous interest is not often used in financial calculations. (Daily compounding, $(1 + r/365)^365$, is generous enough for your bank account, thank you very much. But seriously, daily compounding is a pretty good approximation of continuous growth.)

The exponential e is the bridge from our jumpy “delayed” growth to the smooth changes of the natural world.

A Few Examples

Let’s try a few examples to make sure it’s sunk in. Remember: the APR is the rate they give you, the APY is what you actually earn (your true return).

  • Is a 4.5 APY better than a 4.4 APR, compounded quarterly? You need to compare APY to APY. 4.4% compounded quarterly is $(1 + 4.4\%/4)^4 = 4.47% $, so the 4.5% APY is still better.
  • Should I pay my mortgage at the end of the month, or the beginning? The beginning, for sure. This way you knock out a chunk of debt early, preventing that “debt factory” from earning interest for 30 days. Suppose your loan APY is 6% and your monthly payment is \$2000. By paying at the start of the month, you’d save \$2000 * 6% = \$120/year, or \$3600 throughout a 30-year mortgage. And a few grand is nothing to sneeze at.
  • Should I use several small payments, or one large payment?. You want to pay debt off as early as possible. \$500/week for 4 weeks is better than \$2000 at the end of the month. Each payment stops a few weeks’ worth of interest. The math is a bit tricker, but think of it as 4 \$500 investments, each getting different return. In a month, the first payment saves 3 week’s worth of interest: $500 * (1 + daily rate)^21$. The next saves 2 weeks: $500 * (1 + daily rate)^14$. The third saves a week $500 * (1 + daily rate)^7$ and the last payment doesn’t save any interest. Regardless of the details, prepayment will save you money.

The general principle: When investing, get interest paid early, so it can compound. When borrowing, pay debt early to prevent that interest from compounding.

Onward and Upward

This is a lot for one sitting, but I hope you’ve seen the big picture:

  • The interest rate (APR) is the “speed” at which money grows.
  • Compounding lets you adjust your “speed” as you earn more interest. The APR is the initial speed; the APY is the actual change during the year.
  • Man-made growth uses $(1+r)^n$, or some variant. We like our loans to line up with years.
  • Nature uses $e^{rt}$. The universe doesn’t particularly care for our solar calendar.
  • Interest rates are tricky. When in doubt, ask for the APY and pay debt early.

Treating interest in this funky way (trajectories and factories) will help us understand some of e’s cooler properties, which come in handy for calculus. Also, try the Rule of 72 for a quick way to compute the effect of interest rates mentally (that investment with 6% APY will double in 12 years). Happy math.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Demystifying the Natural Logarithm (ln)

After understanding the exponential function, our next target is the natural logarithm.

Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of $e^x$, a strange enough exponent already.

But there’s a fresh, intuitive explanation: The natural log gives you the time needed to reach a certain level of growth.

natural log time vs growth

Suppose you have an investment in gummy bears (who doesn’t?) with an interest rate of 100% per year, growing continuously. If you want 10x growth, assuming continuous compounding, you’d wait only $\ln(10)$ or 2.302 years. Don’t see why it only takes a few years to get 10x growth? Don’t see why the pattern is not 1, 2, 4, 8? Read more about e.

e and the Natural Log are twins:

  • $e^x$ is the amount we have after starting at 1.0 and growing continuously for $x$ units of time
  • $\ln(x)$ (Natural Logarithm) is the time to reach amount $x$, assuming we grew continuously from 1.0

Not too bad, right? While the mathematicians scramble to give you the long, technical explanation, let’s dive into the intuitive one.

E is About Growth

The number e is about continuous growth. As we saw last time, $e^x$ lets us merge rate and time: 3 years at 100% growth is the same as 1 year at 300% growth, when continuously compounded.

We can take any combination of rate and time (50% for 4 years) and convert the rate to 100% for convenience (giving us 100% for 2 years). By converting to a rate of 100%, we only have to think about the time component:

\displaystyle{e^x = e^{\text{rate} \cdot \text{time}} = e^{1.0 \cdot \text{time}} = e^{\text{time}}}

Intuitively, $e^x$ means:

  • How much growth do I get after after x units of time (and 100% continuous growth)
  • For example: after 3 time periods I have $e^3$ = 20.08 times the amount of “stuff”.

$e^x$ is a scaling factor, showing us how much growth we’d get after $x$ units of time.

Natural Log is About Time

The natural log is the inverse of $e^x$, a fancy term for opposite. Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln.

Now what does this inverse or opposite stuff mean?

  • $e^x$ lets us plug in time and get growth.
  • $\ln(x)$ lets us plug in growth and get the time it would take.

For example:

  • $e^3$ is 20.08. After 3 units of time, we end up with 20.08 times what we started with.
  • $\ln(20.08)$ is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate).

With me? The natural log gives us the time needed to hit our desired growth.

Logarithmic Arithmetic Is Not Normal

You’ve studied logs before, and they were strange beasts. How’d they turn multiplication into addition? Division into subtraction? Let’s see.

What is $\ln(1)$? Intuitively, the question is: How long do I wait to get 1x my current amount?

Zero. Zip. Nada. You’re already at 1x your current amount! It doesn’t take any time to grow from 1 to 1.

  • $\ln(1) = 0$

Ok, how about a fractional value? How long to get 1/2 my current amount? Assuming you are growing continuously at 100%, we know that $\ln(2)$ is the amount of time to double. If we reverse it (i.e., take the negative time) we’d have half of our current value.

  • $\ln(.5) = – \ln(2) = -.693$

Makes sense, right? If we go backwards .693 units (negative seconds, let's say) we’d have half our current amount. In general, you can flip the fraction and take the negative: $\ln(1/3) = – \ln(3) = -1.09$. This means if we go back 1.09 units of time, we’d have a third of what we have now.

Ok, how about the natural log of a negative number? How much time does it take to “grow” your bacteria colony from 1 to -3?

It’s impossible! You can’t have a “negative” amount of bacteria, can you? At most (er… least) you can have zero, but there’s no way to have a negative amount of the little critters. Negative bacteria just doesn’t make sense.

  • $\ln(\text{negative number}) = \text{undefined}$

Undefined just means “there is no amount of time you can wait” to get a negative amount. (Well, if we use imaginary exponentials, there is a solution. But today let's keep it real.)

Logarithmic Multiplication is Mighty Fun

How long does it take to grow 9x your current amount? Sure, we could just use ln(9). But that’s too easy, let’s be different.

We can consider 9x growth as tripling (taking $\ln(3)$ units of time) and then tripling again (taking another $\ln(3)$ units of time):

  • Time to grow 9x = $\ln(9)$ = Time to triple and triple again = $\ln(3) + \ln(3)$

Interesting. Any growth number, like 20, can be considered 2x growth followed by 10x growth. Or 4x growth followed by 5x growth. Or 3x growth followed by 6.666x growth. See the pattern?

  • $\ln(a*b) = \ln(a) + \ln(b)$

The log of a times b = log(a) + log(b). This relationship makes sense when you think in terms of time to grow.

If we want to grow 30x, we can wait $\ln(30)$ all at once, or simply wait $\ln(3)$, to triple, then wait $\ln(10)$, to grow 10x again. The net effect is the same, so the net time should be the same too (and it is).

How about division? $\ln(5/3)$ means: How long does it take to grow 5 times and then take 1/3 of that?

Well, growing 5 times is $\ln(5)$. Growing 1/3 is $-\ln(3)$ units of time. So

  • $\ln(5/3) = \ln(5) – \ln(3)$

Which says: Grow 5 times and “go back in time” until you have a third of that amount, so you’re left with 5/3 growth. In general we have

  • $\ln(a/b) = \ln(a) – \ln(b)$

I hope the strange math of logarithms is starting to make sense: multiplication of growth becomes addition of time, division of growth becomes subtraction of time. Don’t memorize the rules, understand them.

Using Natural Logs With Any Rate

“Sure,” you say, “This log stuff works for 100% growth but what about the 5% I normally get?”

It’s no problem. The “time” we get back from $\ln()$ is actually a combination of rate and time, the “x” from our $e^x$ equation. We just assume 100% to make it simple, but we can use other numbers.

Suppose we want 30x growth: plug in $\ln(30)$ and get 3.4. This means:

  • $e^x = \text{growth}$
  • $e^{3.4} = 30$

And intuitively this equation means “100% return for 3.4 years is 30x growth”. We can consider the equation to be:

\displaystyle{e^{x} = e^{\text{rate} \cdot \text{time}}}

\displaystyle{e^{100 \% \cdot 3.4 \text{years}} = 30}

We can modify “rate” and “time”, as long as rate * time = 3.4. For example, suppose we want 30x growth — how long do we wait assuming 5% return?

  • $\ln(30) = 3.4$
  • $\text{rate} * \text{time} = 3.4$
  • $.05 * \text{time} = 3.4$
  • $\text{time} = 3.4 / .05 = 68 \text{years}$

Intuitively, I think "$\ln(30) = 3.4$, so at 100% growth it will take 3.4 years. If I double the rate of growth, I halve the time needed."

  • 100% for 3.4 years = 1.0 * 3.4 = 3.4
  • 200% for 1.7 years = 2.0 * 1.7 = 3.4 [200% growth means half the time]
  • 50% for 6.8 years = 0.5 * 6.8 = 3.4 [50% growth means double the time]
  • 5% for 68 years = .05 * 68 = 3.4 [5% growth means 20x the time]

Cool, eh? The natural log can be used with any interest rate or time as long as their product is the same. You can wiggle the variables all you want.

Awesome example: The Rule of 72

The Rule of 72 is a mental math shortcut to estimate the time needed to double your money. We’re going to derive it (yay!) and even better, we’re going to understand it intuitively.

How long does it take to double your money at 100% interest, compounded every year?

Uh oh. We’ve been using natural log for continuous rates, but now you’re asking for yearly interest? Won’t this mess up our formula? Yes, it will, but at reasonable interest rates like 5%, 6% or even 15%, there isn’t much difference between yearly compounded and fully continuous interest. So the rough formula works, uh, roughly and we’ll pretend we’re getting fully continuous interest.

Now the question is easy: How long to double at 100% interest? ln(2) = .693. It takes .693 units of time (years, in this case) to double your money with continuous compounding with a rate of 100%.

Ok, what if our interest isn’t 100% What if it’s 5% or 10%?

Simple. As long as rate * time = .693, we’ll double our money:

  • rate * time = .693
  • time = .693/rate

So, if we only had 10% growth, it’d take .693 / .10 or 6.93 years to double.

To simplify things, let’s multiply by 100 so we can talk about 10 rather than .10:

  • time to double = 69.3/rate, where rate is assumed to be in percent.

Now the time to double at 5% growth is 69.3/5 or 13.86 years. However, 69.3 isn’t the most divisible number. Let’s pick a close neighbor, 72, which can be divided by 2, 3, 4, 6, 8 and many more numbers.

  • time to double = 72/rate

which is the rule of 72! Easy breezy.

If you want to find the time to triple, you’d use ln(3) ~ 109.8 and get

  • time to triple = 110 / rate

Which is another useful rule of thumb. The Rule of 72 is useful for interest rates, population growth, bacteria cultures, and anything that grows exponentially.

Where to from here?

I hope the natural log makes more sense — it tells you the time needed for any amount of exponential growth. I consider it “natural” because e is the universal rate of growth, so ln could be considered the “universal” way to figure out how long things take to grow.

When you see $\ln(x)$, just think “the amount of time to grow to x”. In the next article we’ll bring e and ln together, and the sweet aroma of math will fill the air.

Appendix: The Natural Log of E

Quick quiz: What’s $\ln(e)$?

  • The math robot says: Because they are defined to be inverse functions, clearly $\ln(e) = 1$
  • The intuitive human: ln(e) is the amount of time it takes to get “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so $\ln(e) = 1$.

Think intuitively.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

An Intuitive Guide To Exponential Functions & e

e has always bothered me — not the letter, but the mathematical constant. What does it really mean?

Math books and even my beloved Wikipedia describe e using obtuse jargon:

The mathematical constant e is the base of the natural logarithm.

And when you look up the natural logarithm you get:

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

Nice circular reference there. It’s like a dictionary that defines labyrinthine with Byzantine: it’s correct but not helpful. What’s wrong with everyday words like “complicated”?

I’m not picking on Wikipedia — many math explanations are dry and formal in their quest for rigor. But this doesn’t help beginners trying to get a handle on a subject (and we were all a beginner at one point).

No more! Today I’m sharing my intuitive, high-level insights about what e is and why it rocks. Save your rigorous math book for another time. Here’s a quick video overview of the insights:

e is NOT Just a Number

Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point.

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.

Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.

Understanding Exponential Growth

Let's start by looking at a basic system that doubles after an amount of time. For example,

  • Bacteria can split and “doubles” every 24 hours
  • We get twice as many noodles when we fold them in half.
  • Your money doubles every year if you get 100% return (lucky!)

And it looks like this:

2 times growth

Splitting in two or doubling is a very common progression. Sure, we can triple or quadruple, but doubling is convenient, so hang with me here.

Mathematically, if we have x splits then we get $2^x$ times as much stuff than when we started. With 1 split we have $2^1$ or 2 times as much. With 4 splits we have $2^4 = 16$ times as much. As a general formula:

\displaystyle{ \text{growth} = 2^x}

Said another way, doubling is 100% growth. We can rewrite our formula like this:

\displaystyle{ \text{growth} = (1 + 100\%)^x}

It’s the same equation, but we separate 2 into what it really is: the original value (1) plus 100%. Clever, eh?

Of course, we can substitute any number (50%, 25%, 200%) for 100% and get the growth formula for that new rate. So the general formula for x periods of return is:

\displaystyle{\text{growth} = (1 + \text{return})^x}

This just means we use our rate of return, (1 + return), “x” times.

A Closer Look

Our formula assumes growth happens in discrete steps. Our bacteria are waiting, waiting, and then boom, they double at the very last minute. Our interest earnings magically appear at the 1 year mark. Based on the formula above, growth is punctuated and happens instantly. The green dots suddenly appear.

The world isn’t always like this. If we zoom in, we see that our bacterial friends split over time:

2 times growth detail

Mr. Green doesn’t just show up: he slowly grows out of Mr. Blue. After 1 unit of time (24 hours in our case), Mr. Green is complete. He then becomes a mature blue cell and can create new green cells of his own.

Does this information change our equation?

Nope. In the bacteria case, the half-formed green cells still can’t do anything until they are fully grown and separated from their blue parents. The equation still holds.

Money Changes Everything

But money is different. As soon as we earn a penny of interest, that penny can start earning micro-pennies of its own. We don’t need to wait until we earn a complete dollar in interest — fresh money doesn’t need to mature.

Based on our old formula, interest growth looks like this:

interest rate growth

But again, this isn’t quite right: all the interest appears on the last day. Let’s zoom in and split the year into two chunks. We earn 100% interest every year, or 50% every 6 months. So, we earn 50 cents the first 6 months and another 50 cents in the last half of the year:

interest rate 6 months

But this still isn’t right! Sure, our original dollar (Mr. Blue) earns a dollar over the course of a year. But after 6 months we had a 50-cent piece, ready to go, that we neglected! That 50 cents could have earned money on its own:

compound interest

Because our rate is 50% per half year, that 50 cents would have earned 25 cents (50% times 50 cents). At the end of 1 year we’d have

  • Our original dollar (Mr. Blue)
  • The dollar Mr. Blue made (Mr. Green)
  • The 25 cents Mr. Green made (Mr. Red)

Giving us a total of \$2.25. We gained \$1.25 from our initial dollar, even better than doubling!

Let’s turn our return into a formula. The growth of two half-periods of 50% is:

\displaystyle{\text{growth} = (1 + 100\%/2)^{2} = 2.25}

Diving into Compound Growth

It’s time to step it up a notch. Instead of splitting growth into two periods of 50% increase, let’s split it into 3 segments of 33% growth. Who says we have to wait for 6 months before we start getting interest? Let’s get more granular in our counting.

Charting our growth for 3 compounded periods gives a funny picture:

4  month compound interest

Think of each color as shoveling money upwards towards the other colors (its children), at 33% per period:

  • Month 0: We start with Mr. Blue at \$1.
  • Month 4: Mr. Blue has earned 1/3 dollar on himself, and creates Mr. Green, shoveling along 33 cents.
  • Month 8: Mr. Blue earns another 33 cents and gives it to Mr. Green, bringing Mr. Green up to 66 cents. Mr. Green has actually earned 33% on his previous value, creating 11 cents (33% * 33 cents). This 11 cents becomes Mr. Red.
  • Month 12: Things get a bit crazy. Mr. Blue earns another 33 cents and shovels it to Mr. Green, bringing Mr. Green to a full dollar. Mr. Green earns 33% return on his Month 8 value (66 cents), earning 22 cents. This 22 cents gets added to Mr. Red, who now totals 33 cents. And Mr. Red, who started at 11 cents, has earned 4 cents (33% * .11) on his own, creating Mr. Purple.

Phew! The final value after 12 months is: 1 + 1 + .33 + .04 or about 2.37.

Take some time to really understand what’s happening with this growth:

  • Each color earns interest on itself and hands it off to another color. The newly-created money can earn money of its own, and on the cycle goes.
  • I like to think of the original amount (Mr. Blue) as never changing. Mr. Blue shovels money to create Mr. Green, a steady 33 every 4 months since Mr. Blue does not change. In the diagram, Mr. Blue has a blue arrow showing how he feeds Mr. Green.
  • Mr. Green just happens to create and feed Mr. Red (green arrow), but Mr. Blue isn’t aware of this.
  • As Mr. Green grows over time (being constantly fed by Mr. Blue), he contributes more and more to Mr. Red. Between months 4-8 Mr. Green gives 11 cents to Mr. Red. Between months 8-12 Mr. Green gives 22 cents to Mr. Red, since Mr. Green was at 66 cents during Month 8. If we expanded the chart, Mr. Green would give 33 cents to Mr. Red, since Mr. Green reached a full dollar by Month 12.

Make sense? It’s tough at first — I even confused myself a bit while putting the charts together. But see that each dollar creates little helpers, who in turn create helpers, and so on.

We get a formula by using 3 periods in our growth equation:

\displaystyle{\text{growth} = (1 + 100\%/3)^3 = 2.37037...}

We earned \$1.37, even better than the \$1.25 we got last time!

Can We Get Infinite Money?

Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket?

Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:

n          (1 + 1/n)^n
-----------------------
1          2
2          2.25
3          2.37
5          2.488
10         2.5937
100        2.7048
1,000      2.7169
10,000     2.71814
100,000    2.718268
1,000,000  2.7182804
...

The numbers get bigger and converge around 2.718. Hey… wait a minute… that looks like e!

Yowza. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:

\displaystyle{\text{perfect compound growth} = e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n}

This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2.718.

But what does it all mean?

The number e (2.718…) is the maximum possible result when compounding 100% growth for one time period. Sure, you started out expecting to grow from 1 to 2 (that’s a 100% increase, right?). But with each tiny step forward you create a little dividend that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2. e is the maximum, what happens when we compound 100% as much as possible.

So, if we start with \$1.00 and compound continuously at 100% return we get 1e. If we start with \$2.00, we get 2e. If we start with \$11.79, we get 11.79e.

e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process. You might not always reach the speed limit, but it’s a reference point: you can write every rate of growth in terms of this universal constant.

Aside:

  • We want to clarify the difference between the growth and the final result:

    e (final result, ~2.718...) = original (1.0) + growth (1.718...)

  • The growth can be further split into "direct growth" (Mr. Green) and "growth on growth" (the other colors):

    e (final result, ~2.718) = original (1.0) + direct growth (1.0) + compound growth (.718...)

What about different rates?

Good question. What if we grow at 50% annually, instead of 100%? Can we still use e?

Let’s see. The rate of 50% compound growth would look like this:

\displaystyle{\lim_{n\to\infty} \left( 1 + \frac{.50}{n} \right)^n}

Hrm. What can we do here? Remember, 50% is the total return, and n is the number of periods to split the growth into for compounding. If we pick n=50, we can split our growth into 50 chunks of 1% interest:

\displaystyle{\left( 1 + \frac{.50}{50} \right)^{50} = \left( 1 + .01 \right)^{50}}

Sure, it’s not infinity, but it’s pretty granular. Now imagine we also divided our regular rate of 100% into chunks of 1%:

\displaystyle{e \approx \left( 1 + \frac{1.00}{100} \right)^{100} = \left( 1 + .01 \right)^{100}}

Ah, something is emerging here. In our regular case, we have 100 cumulative changes of 1% each. In the 50% scenario, we have 50 cumulative changes of 1% each.

Different exponential rates

What is the difference between the two numbers? Well, it’s just half the number of changes:

\displaystyle{\left( 1 + .01 \right)^{50} = \left( 1 + .01 \right)^{100/2} = \left( \left( 1 + .01 \right)^{100}\right)^{1/2} = e^{1/2} }

This is pretty interesting. 50 / 100 = .5, which is the exponent we raise e to. This works in general: if we had a 300% growth rate, we could break it into 300 chunks of 1% growth. This would be triple the normal amount for a net rate of $e^3$.

Even though growth can look like addition (+1%), we need to remember that it’s really a multiplication (x 1.01). This is why we use exponents (repeated multiplication) and square roots ($e^{1/2}$ means “half” the number of changes, i.e. half the number of multiplications).

Although we picked 1%, we could have chosen any small unit of growth (.1%, .0001%, or even an infinitely small amount!). The key is that for any rate we pick, it’s just a new exponent on e:

\displaystyle{\text{growth} = e^{\text{rate}}}

What about different times?

Suppose we have 300% growth for 2 years. We’d multiply one year’s growth ($e^3$) by itself:

\displaystyle{\text{growth} = \left(e^{3}\right)^{2} = e^{6}}

And in general:

\displaystyle{\text{growth} = \left(e^{\text{rate}}\right)^{\text{time}} = e^{\text{rate} \cdot \text{time}}}

Because of the magic of exponents, we can avoid having two powers and just multiply rate and time together in a single exponent.

The big secret: e merges rate and time.

This is wild! $e^x$ can mean two things:

  • x is the number of times we multiply a growth rate: 100% growth for 3 years is $e^3$
  • x is the growth rate itself: 300% growth for one year is $e^3$

Won’t this overlap confuse things? Will our formulas break and the world come to an end?

It all works out. When we write $e^x$, the variable $x $is a combination of rate and time.

\displaystyle{x = \text{rate} \cdot \text{time}}

Let me explain. When dealing with continuous compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).

  • 10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year.
  • 1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop.

The same “30 changes of 1%” happen in each case. The faster your rate (30%) the less time you need to grow for the same effect (1 year). The slower your rate (3%) the longer you need to grow (10 years).

But in both cases, the growth is $e^{.30} = 1.35$ in the end. We’re impatient and prefer large, fast growth to slow, long growth but e shows they have the same net effect.

So, our general formula becomes:

\displaystyle{\text{growth} = e^x = e^{rt}}

If we have a return of r for t time periods, our net compound growth is $e^{rt}$. This even works for negative and fractional returns, by the way.

Example Time!

Examples make everything more fun. A quick note: We’re so used to formulas like $2^x$ and regular, compound interest that it’s easy to get confused (myself included). Read more about simple, compound and continuous growth.

These examples focus on smooth, continuous growth, not the jumpy growth that happens at yearly intervals. There are ways to convert between them, but we’ll save that for another article.

Example 1: Growing crystals

Suppose I have 300kg of magic crystals. They’re magic because they grow throughout the day: I watch a single crystal, and in the course of 24 hours it sheds off its own weight in crystals. (The baby crystals start growing immediately at the same rate, but I can’t track that — I’m watching how much the original sheds). How much will I have after 10 days?

Well, since the crystals start growing immediately, we want continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: $300 \cdot e^{1 \cdot 10} = 6.6 \text{million kg}$ of our magic gem.

This can be tricky: notice the difference between the input rate and the total output rate. The “input” rate is how much a single crystal changes: 100% in 24 hours. The net output rate is e (2.718x) because the baby crystals grow on their own.

In this case we have the input rate (how fast one crystal grows) and want the total result after compounding (how fast the entire group grows because of the baby crystals). If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log.

Example 2: Maximum interest rates

Suppose I have \$120 in an account with 5% interest. My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years?

Our rate is 5%, and we’re lucky enough to compound continuously. After 10 years, we get $120 \cdot e^{.05 \cdot 10} = 197.85$. Of course, most banks aren’t nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don’t like you.

Example 3: Radioactive decay

I have 10kg of a radioactive material, which appears to continuously decay at a rate of 100% per year. How much will I have after 3 years?

Zip? Zero? Nothing? Think again.

Decaying continuously at 100% per year is the trajectory we start off with. Yes, we do begin with 10kg and expect to “lose it all” by the end of the year, since we’re decaying at 10 kg/year.

We go a few months and get to 5kg. Half a year left? Nope! Now we’re losing at a rate of 5kg/year, so we have another full year from this moment!

We wait a few more months, and get to 2kg. And of course, now we’re decaying at a rate of 2kg/year, so we have a full year (from this moment). We get 1 kg, have a full year, get to .5 kg, have a full year — see the pattern?

As time goes on, we lose material, but our rate of decay slows down. This constantly changing growth is the essence of continuous growth & decay.

After 3 years, we’ll have $10 \cdot e^{(-1)(3)} = 10e^{-3} = .498 \text{ kg}$. We use a negative exponent for decay — we want a fraction ($1/e^{rt}$) vs a growth multiplier ($e^{rt}$). [Decay is commonly given in terms of "half life" -- we'll talk about converting these rates in a future article.]

More Examples

If you want fancier examples, try the Black-Scholes option formula (notice e used for exponential decay in value) or radioactive decay. The goal is to see $e^{rt}$ in a formula and understand why it’s there: it’s modeling a type of growth or decay.

And now you know why it’s “e”, and not pi or some other number: e raised to “r*t” gives you the growth impact of rate r and time t.

There’s More To Learn

My goal was to:

  • Explain why e is important: It’s a fundamental constant, like pi, that shows up in growth rates.
  • Give an intuitive explanation: e lets you see the impact of any growth rate. Every new “piece” (Mr. Green, Mr. Red, etc.) helps add to the total growth.
  • Show how it’s used: $e^x$ lets you predict the impact of any growth rate and time period.
  • Get you hungry for more: In the upcoming articles, I’ll dive into other properties of e.

This article is just the start — cramming everything into a single page would tire you and me both. Dust yourself off, take a break and learn about e’s evil twin, the natural logarithm.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.