@Anonymous: I know what you mean! It took me a long, long time to really understand e / ln to a level that made intuitive sense!
how would you solve 8e^9t=12e^8t ?
Man, I do not even know where to begin. I love your site and articles. e and ln articles are incredible, real art. In 5 min you have answered some questions that bugged me for years. You have amazing talent in teaching and I would love to see math/calculus book written by you some day. Engineers, computer science and math enthusiasts do not need “rigor”, they need real knowledge and understanding of the subject. And honestly I have never read anything even closely as good as what you have put on this site. Please keep it up!!!
You didn’t mention that ln(0) is also undefined, not just ln(negative number).
But great site! Now I can go teach my teacher!
Great Job, Thanks
Wonderful article.
In response to comments 48 and 49: This is a little technical, but it’s a really neat point, and maybe (just maybe) it’ll get some more people interested in math. If you allow yourself to wade into the world of complex numbers, taking the logarithm with a negative base yields a very interesting result. As you said, log(base -2) only maps onto the integers for certain cases; for example -2^0=1, -2^1=-2, -2^2=4, etc. But for -2^0.5, you get isqrt(2), a purely imaginary number. Continuing, -2^1.5=(-1)^(1.5)2^(1.5)=-isqrt(8). Now, if we plot these in the complex plane, we have a point on the real axis at 1, a point on the imaginary axis at 1.414, a point on the real axis at -2, a point on the imaginary axis at -2.828, and a point on the real axis at 4. The kicker now is to figure out how to handle a number like -2^(1/3). This equation basically asks you to look for a number that, when cubed, gives you -2. You can check that 1/2 + isqrt(3)/2 does just this. This gives you a new point on the graph at 0.5 on the real axis and 0.866 on the imaginary axis. If you haven’t tuned out yet, you’re probably starting to see a pattern here. In fact, if you keep graphing points like this, you get a figure known as a logarithmic spiral. This type of spiral is seen all over the place in nature. Wikipedia gives great pictures of the spiral arms of a galaxy, the inspiral of a hurricane, and a nautilus shell to name a few.
Wow, im a student at syracuse univeristy and I have the worlds worst precalc prof. and thoroughly dimishing my interest in this class, however, I cant give up cuz Im spending 50,000 to go here and plus im premed. but let me just say if my prof could get up off his high horse and take the stick out of his rear behind and teach logs like this or anthing like this, I would actually be learning something and wouldnt have to resort to going online, when Im paying this much. thank you sir/ma’am. this has helped enormously! 
Hello, I would like to say that this post has helped me enormously,
I go to Syracuse University and I have the worlds worst precalc teacher. If he could get up of his high horse and take that stick out of his rear behind, then he could actually succeed at being a teacher. Why dont people teach you the basics, stop with the formulas and TEACH people. well thanks so much sir/ ma’am.
WOW! Thank you Kalid!
I’m a university student in New Zealand, and I only took maths for the first three years of high school. Now that I’m back studying I’ve realised that calculus wasn’t just a series of classes that I bunked in high school maths, but a subject that I might (shock, horror) actually need it for my Economics degree.
I stumbled across your articles while I was quite literally in tears about the idea of learning calculus - I love economics but I was even considering changing my major because it just seemed too hard, and my head hurts the moment I look at a formula. I’ve been reading various stuff on your site for a solid hour and my head doesn’t hurt a bit! I intend to send a link to my lecturer in the hope that he will recommend this site to all the other social science (rather than commerce) students who are struggling to get their heads around the maths of economics,the most beautiful subject in the world.
@Izzy: Awesome! Really, really glad it helped! I completely know what you mean about the feeling of frustration – it seems math evokes it more often than other subjects :). I’m thrilled you stuck with it and found value in the articles here – thanks for sharing it!
How do you estimate “negative feedback” like the slow down of growth in bacteria when they run out of media. Or when the growth of a real estate development runs out of attractive sites. My guess is you need some parallel process and use empirical corrections.
" * 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"
Aren’t you scrambling up two different concepts here? e^x is not the amount of “continuous growth” – it is the AMOUNT, period. x = rate * time. One of the first things I learned in elementary algebra is that rate * time = distance (“amount” is just another term for “distance”). e^x is the AMOUNT accrued “after a certain amount of time”.
Conversely, ln is not “the amount of time needed to reach a certain level of continuous growth” – it is the amount of time needed to reach a certain amount.
@Robert: Great question, I need to think about how to clarify this. If someone says “I’m growing at 100%, how much will I have after 3 time periods?”, there’s a few answers:
- e^3 if you are growing continuously
- 2^3 if you compound every time period
- 1 + 1 + 1 + 1 if you assume simple interest (no compounding)
While all answers are some type of “amount”, I find it helpful to qualify how that answer was determined (did we assume continuous, compound, or simple interest?). There may be a better way to phrase it, but the key is to understand how the amount was computed.
Hi. I will read more of this article today (just been reading your article on e - it’s truly excellent). I see someone else has mentioned the Gibbs free energy equation here. Is there any chance you could explain why the natural logarithm is used there??? The “RTlnK” bit. The natural log of the concentration of products over the reactants ln[products]/[reactants]? I don’t understand why that is done. Why, why, why? Someone please help 
I will worship you forever.
"If the population of Wyoming is changing exponentially and in 1980 the population of Wyoming was 469,557 and in 1990 it was 453,588 and in 2000 the population was 493,782. What function P(t)= P*e^kt is suggested for the given data for 1980 and 1990?"
Can you please show me how to answer this problem it is really confusing me, because the rate seems to change from negative to positive.
When I first read this article, it seemed to make sense. But then I thought I would just go over it again and now I see something that doesn’t make sense to me (perhaps because I’ve taken it out of context). You wrote:
“100% return for 3.4 years is 30x growth.”
Say each year on New Year’s Day, I place a bet at 1:1 odds, which means I earn 100% return on my money. If I start with $1 in year one, then on Jan. 1 of year two I now have $2. On Jan 1 of year three I have $4, and on Jan 1 of year four I have $8. That’s not 30 times growth.
If a rate of return is defined as 100% per period, then it’s 100%. It’s not 100% compounded continuously. 100% per year, compounded continuously, is 271.8% (or so) per year.
So intuitively, I’m having trouble arbitrarily converting growth rates to some power of e that in effect, change the rules in the middle of the game. It’s more intuitive to me to just say that e to the 3.4 power is 30, and that ln 30 is 3.4. I really do appreciate the idea of intuitive and imaginative explanations, and I actually really dig your site and think your the entire concept is really bold and timely. This article just didn’t work for me, though.
Logarithms are just high powered “demagnifiers” that let us work with really large numbers over time on a human scale. The natural log of a number is just that number’s value as a power of e, and that works for certain types of models and not others. It gives you the time needed to reach a certain level of growth but ONLY if that growth is modeled by e, or it could be used to find the growth rate as a power of e given a particular time frame. I hope I got that right! 
So e is an interesting number but so is 360, right? We could just define e as 10^.4342944819 and call it “abracadabra” and maybe that’s just what’s needed to demythologize it and make working with it more intuitive.
@Fred: Great questions!
For the doubling example, it’s important to recognize there are two rates: the “input” rate and the final “output” rate. In the betting case you describe, we agree to only look at the total output (doubling) at perfect year-long intervals. For most man-made things, this is simple and works.
For most natural processes, however, we only know the current/instantaneous rate: i.e., I’m constantly growing at a rate of 50%, where does that put me in a year? It won’t be 50% higher because in 1 minute I’ll have grown a bit and will then be growing at 50% of that new rate. e lets us work out what the final output will be.
Unfortunately, “interest rate” is an overloaded term. In the banking industry, they talk about APR (annual percentage rate) and APY (annual percentage yield) - APR is the input, APY is the output (what you actually pay). Simple, compound, or continuous growth patterns can take the same APR (of 10%, say) and return a different APY (total result after the time period).
Yes, I like the note about “demagnifiers” – the natural log lets us “undo” the effect of exponential growth. But here’s the tricky thing: any amount of growth can be represented by e! Even processes that didn’t start growing continuously (like your doubling) can be described by an equivalent continuous process. That is, even though 3 * 4 = 12 isn’t a square number, 12 can still be seen as sqrt(12) * sqrt(12). The neat thing about continuous growth/ln/e is that it’s a standard reference point to compare different growth rates.
e is interesting because it’s the basis for this standard reference point. Indeed, we could say e = 2.71828… and memorize it (which is often done!) but it’s more illuminating to say “e is the number when you perfectly compound 100% growth for 1 time period”, just like “pi is the circumference of a circle of diameter 1”. Just giving the raw result can hide the reasoning behind why the number is so useful.
Hope this helps!
Thanks Kalid. BTW, in my example I miscalculated the effect of continuous compounding as “271.828% growth.” I neglected to subtract the principal when calculating the percentage increase. A stated rate of 100% return over a period, compounded continuously actually represents 171.828% effective growth at the end of that period. (Oops.)
The contradiction of course is that if the stated rate has a major asterisk, “compounded continuously”, then why even bother quoting the stated rate? If I borrow at 10% per year, there had better not be any fine print that says “compounded continuously.” That’s why banks are required to state the APR on a loan and not just the interest rate. Ignoring fees, the real interest rate paid is determined based on a combination of the stated (simple) rate and the method of compounding.
Of course I was downplaying the importance of e. e^x is the only function that equals its own derivative, and it’s essential for computing probabilities. It’s spooky the way it keeps popping up in all sorts of real-world situations, and probably will for a very long time.
I like the idea you presented in a prior article that “e represents the idea that all continually growing systems are scaled versions of a common rate.” I also like the way you point out here that the natural log’s value is composed of the factors of rate and time, a very helpful way of thinking about continuous money growth.
Also: I do see what you’re saying about the natural log being a useful benchmark for comparing different growth rates, whether continuous or not.
thanks.