Demystifying the Natural Logarithm (ln)

Hi Rhonda, I’m so happy you found the article useful! Yes, there are many concepts that just get thrown out there without the background about why it’s useful.

e/ln aren’t magical, just the result of wondering what would happen if you earned interest as fast as you could (i.e. compounding every fraction of a second). Good luck with your teaching!

Does this make sense? Mike Y said all that about simple vs continuous, and how could it be continuous if it’s simple. But think of the bacteria: If one bacterium can split in one hour, it will have a growth rate of 2, not e, because half of the bacterium cannot start splitting until it is completely split right? Wouldn’t that be the same for populations, so e only applies to things that grow continuously.

I guess I’m wondering, how could a population increase by 1.5, wouldn’t all the bacteria take the entire day to split, or if each were offset, you couldn’t count the incomplete bacteria in the total.

The thing is, if bacteria divided more than once per day, the rate would be greater than 100% per day, but they can’t divide continuously regardless, the time it takes a bacteria to divide would be the time it takes a population to double. I don’t see how this relates to continuous compounding

oh your article on e explains the difference between bacteria and money

can someone tel me how to solve the logarithm of a negative number… for example log of -1 to the base 10… i know that the answer is a complex number…

Great question. You can plug in negative logs to google calculator and do ln(-1) / ln (10). So log base 10 of -1 is about 1.36i.

Euler’s Formula explains how to do negative logs, and I’ll cover this in a future article :).

@Trying to understand: Yep, money and bacteria grow differently (simple vs. compound). However, even bacterial growth can be modeled continuous growth at a different rate. And in reality, there are millions of bacteria at different stages, so the total “clump” looks like one continuously changing amount.

This site is amazing. I have a math degree from a fancy school and I never really understood ln until now. Thanks so much!

Thanks Louis! Yes, sometimes it’s easy to miss the real insights when learning a subject at a deep level. Personally, I’m pretty interested in math and am trying to study analysis, hopefully some “aha” insights will click there :).

cheers man, certinly cleared up some stuff for me. Wish I had this a few years ago at school. I think I get the discrete growth thing, but this really helped me with continuous change. Gonna help me in years to come. You should post on Wikipedia, the maths stuff there is inaccessable from a basic level of knowledge, whereas the article here is practical, much to my liking. Keep it up.

First, let me say I think your explanation of ln is very intuitive and helpful.

However, your first example threw me off a bit because your terminology is incorrect with regards to simple interest. I see this is discussed a bit in the comments, but the following comment you made is incorrect: “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.” In the financial world, simple interest means interest on the principal only, with no interest on interest. Compound interest means you get interest on the interest. It can be compounded quarterly, annually, continuously, etc. So constant growth is compound interest, compounded continuously. So, where you say compound interest, I believe you mean continuous compounding. And when you say simple interest, you still mean compounding, but it’s done an an annual or other discrete basis.

For reference see:

@G04T_DFA: Thanks for the comment - glad you enjoyed it! Yeah, I make a few typo edits to Wikipedia every once in a while, but generally save the longer stuff for here :).

@Mike: Thanks for the comment, you’re right: simple interest is like a fixed bond coupon with no reinvestment. As you noticed, I was really talking about the difference between yearly compound interest and continuously compounded interest. I’ll update the article – thanks for the catch!

Thanks for your explanations of compound interest and from whence “e” is derived. Your explanation of compound interest would have saved me from an all night “do-it-by-hand” session 20 years ago. An accountant told me I was correct but it took a very long time to calculate the monies for at least four participants. Henceforth I’m referring a number of readers to you. If there’s anything wrong I’ll have 'em sue you! Keep up the good work.

@N. Kogneato: I appreciate the comment, glad it was helpful! I might need to add a little disclaimer – “Results may vary” :).

Great write up on a useful concept. I work for an insurance company where we use LN to determine the time at which ‘losses’ or ‘investment’ income can/will grow over time. Our actuaries use various formulas that leverage LN and though I took calc in college, I couldn’t figure out the importance of the number. You’ve done what so many people don’t, shown the relevance of math to real world scenarios. Math, Accounting, and other fields are guilty of using jargon and obscure terms without ever explaining the impact of the terms in ‘real world’ tangible ways. I’ve passed this link on to other co-workers who have all found it useful. Thanks.

@Brett: Thanks for the wonderful comment! Yes, I completely agree – we can end up using an idea for years in finance or engineering without understanding what it really means. It just becomes a magic equation we throw numbers into :).

Glad you and your co-workers enjoyed it!

Great explanation for ln, but you already knew that…

I’m working on a problem in my quantum mechanics class, the Baker-Hausdorff lemma, and I can’t figure out how they’re expanding the series! Could you give me a point in the right direction?

z = ln [ (e^A)*(e^B ] = A + B + (1/2)(AB - BA) + (1/12)(A^2B + AB^2 - 2ABA + B^2A + BA^2 - 2BAB)

Is this some sort of twisted Taylor Series expansion?

@BatManda: Thanks – great question. I don’t know much about that formula unfortunately, it does look like a crazy taylor series. It seems strange to break the equation apart when you could simplify it like this:

=ln[ e^(A+B) ]
= A + B

But clearly I must be missing something :). You might try the math/physics forums (there’s a section for quantum):

First, let me say that I found your articles on e and ln to be very enlightening. Thanks for taking the time.

That being said, I agree with the general consensus that your explanation of 100% continuous growth is confusing. I’m not sure if this is valid for all uses of e, but I prefer to think of all cases of continuous growth to be 100% while only time varies. In this way, all talk of percentages can be left behind (as I’m not sure the concept of fractional continuous growth really applies to real world situations) and allows us to simply say “continuous growth.”

Also, I am surprised that no one mentioned the case of continuous growth that everyone who’s taken high school chemistry should know about: dG = -RT ln K which relates the reaction rate constant K to Gibbs Free Energy. Continuous growth is essential to any topic relating to chemistry including phase change, thermodynamics, heat and fluid transport phenomena, and a wide range of other topics. Math is a tool of science, not the other way around.