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**.

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 of continuous growth after a certain amount of time.
- Natural Log (ln) is the amount of
**time**needed to reach a certain level of continuous growth

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

Contents

## 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:

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, 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(negative number) = undefined

Undefined just means “there is no amount of time you can wait” to get a negative amount.

## Logarithmic Multiplication is Mighty Fun

How long does it take to grow 4x your current amount? Sure, we could just use ln(4). But that’s too easy, let’s be different.

We can consider 4x growth as doubling (taking ln(2) units of time) and then doubling again (taking another ln(2) units of time):

- Time to grow 4x = ln(4) = Time to double and double again = ln(2) + ln(2)

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 = 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:

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
- rate * time = 3.4
- .05 * time = 3.4
- time = 3.4 / .05 = 68 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.

## Leave a Reply

306 Comments on "Demystifying the Natural Logarithm (ln)"

Ah, that’s the magic of continuous compound growth. Consider how much bacteria you have at the 12-hour mark: 1.5 (50% more than you started).

After another 12 hours that amount (1.5) has time to grow another 50%: 1.5 * 1.5 = 2.25, even better than double at the end of the day.

Now take a smaller interval, like 6-hours. After 6 hours you have 25% growth: 1.25

and after 4 periods of 25% growth you have 1.25^4 = 2.44 times your original.

If you keep taking smaller and smaller time intervals, you get a net compound growth rate of 2.71828 per day, which is e (take a look at the article on e for more details). I’ll amend the article to clarify.

Perhaps, but that’s NOT what you said, you said they had 100% growth in 1 day. You didn’t say 125% growth in 1 day.

The example you’re giving seems to be based on compounding interest; if you have a 5% interest rate (yearly) BUT you compound it continously you get an APR or whatever of 5.25% or whatever. Basically it works out that ln would be useful IF you wanted to take something that didn’t compound continously, and see what would happen if it did. By definition, a 100% increase in a colony of bacteria in 24 hours means that it will actually have double in size in 24 hours. Yes it would have been growing continously, but the end result would be 100% growth, not e^1.

That’s a good point, I think I’ll rephrase the bacteria example to use money, where the concepts of “simple” and “compound” interest are more natural.

You can imagine bacteria that grows at 100% “simple interest”, and the net result would be over 100% growth if the mini-bacteria which are made after a half-day create new bacteria of their own. But describing it as 100% continuous growth, as worded, may be confusing.

The meta-point, which I’m pretty sure we both agree on, is that ln can give you the time something growing continuously (e^x) would take.

And the cool thing is that ln can even help figure out non-continuous growth (2^x, like the normal bacteria case) as well. I’ll be writing more on this.

I’m not sure that I agree on the “something” grows part. It doesn’t really seem to apply to anything more than interest, or something I can’t think of that might not grow in cleanly divisible increments. As for the ln being useful for the bacteria growth, actually not all. That would be Log base 2 rather than Log base e.

Uh oh, you’re going to make me give away the crown secret: e and ln can be used in a change-of-base formula, so you actually don’t need a separate log base 2 to get the expected number of doublings :) (It’s convenient to have a separate log base 2, but not strictly needed).

excellent explanation…its more intuitive now!

Thanks Raj, glad you liked it!

Wow… i’ve been casually using ln(N) in the context of “big-O notation” in programming for years, but this article is the first to really get my on my way to being able to USE the function. i first tried the related Wikipedia articles and, like you mentioned, was only mystified by the circular references between ln and ‘e’.

Hey Stephan, thanks for the mail! There’s so many topics that we just take for granted and use mechanically, it’s a lot of fun to really understand them. E and natural log was like this for a long time for me, too.

Hi my friend… congratulations for an amazing site! This is the way these things should be taught to children! Will be spreading the word to as many people as I can! Since you’re into e, I wonder if you could write something demystifying a bit hyperbolic trigonometry.

Thanks John, I’m happy you liked the site! I think hyperbolic geometry would be a good topic — personally, I need to investigate it a bit more! But I’d love to share what I learn when I do.

Excellent article.cool..Maths was never so easy to read.

this is very well said;

i don’t know that i’ve ever

seen it put in quite this way.

on the other hand, you’ve confused

the issue mightily by referring

to “simple interest” — which is

essentially the

oppositeof “compound interest” and hence

not what you mean at all.

wikipedia on interest

Hi vlorbik, thanks for the comment and feedback. I’d like to understand what you mean about simple interes vs. compound to help make the article more clear.

For me, I consider continuous interest an extension of simple interest, rather than the opposite. Simple interest has a relatively large term (interest returned over 1 year), while compound interest breaks the duration into many (infinitely small) pieces.

My meaning was that ln(30) means a period of

continuousgrowth for 1n(10) units of time, followed by a period ofcontinuousgrowth for ln(3) units of time.I don’t understand the comment about conversion between simple and compound interest (as described below). If, ln(rt) = ln(2) = .693, then .693…/r should exactly equal the time it takes to double.

Since .693 is approximated by .72, there is some error in this shortcut, but it is not a function of a conversion from compound to simple interest. Please elaborate on your thinking.

One caveat: notice how we’re converting between simple and compound interest – won’t this mess up our formula? Yes, it does, but at reasonable interest rates like 5%, 6% or even 15%, there isn’t much difference between simple and compound interest. So the rough formula works, uh, roughly.

Hi, great question. Simple vs. compound interest is pretty tricky and I want to clarify it in a later article.

There are two sources of error in the rule of 72 equation:

1) The rounding from .693 to .72

2) The use of natural log when most interest is actually simple interest. The natural log (ln) assumes continuous growth, but this is not the case for most returns.

Suppose you have an investment with 5% simple interest.

To compute your return, you get $100 * (1 + .05) = $105 after the first year.

To compute your return with compound interest, you’d have $100 * e^(.05) = $105.13 at the end of the year.

It’s only 13 cents, but it’s a difference that can lead to small errors in how long growth rates will take. (For small rates like 5%, simple vs compound interest isn’t a big deal. For larger rates like 100%, simple interest would be $200, while compound interest would be “e”: $271.281828)

I do need to take another look at all this and make sure my explanation is correct, it can be confusing :)

Your article was very useful and helped me better understand the natural logarithm.

Thanks!

Julien.

Hi Julien, I’m glad you found it useful!

The concept is so clear now! and the explanation is just awesome. Keep up the good work.

Thanks Jason!

This website makes sense of the things that even math teachers have to look up to do correctly.

Amazing job Kalid, absolutely amazing.

Wow, thanks for the kind words! I’m happy to be able to share my thoughts, I’m glad you enjoyed the site :).

[…] The natural log is not just an inverse function. It is about the amount of time things need to grow. […]

e^3 is 20.08. After 3 units of time, we end up with 20.08 more than we started with.

Should that read “we end up with 20.08 times what we started with?”