Demystifying the Natural Logarithm (ln)

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

This is a companion discussion topic for the original entry at

Uhm, wouldn’t bacteria grown at 100% per day take more than 3 days? first day gives you 2x growth, second gives you 4x, third gives you 8x, sometime after 3rd but before 4th gives you 10x. Perhaps .3 days, dunno.
Sorry, I couldn’t read past that seemingly huge error.

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 :slight_smile: (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…Maths was never so easy to read.

ln(a*b) = ln(a) + ln(b)

The log of a times b = log a + log b. This relationship makes sense when you think about it being the 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 ln(10), to grow 10x again. The net effect is the same, so the net time should be the same too (and it is).

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 opposite
of “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 continuous growth for 1n(10) units of time, followed by a period of continuous growth 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 :slight_smile:

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