An Intuitive Guide To Exponential Functions & e

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

This is a companion discussion topic for the original entry at

Nice graphics! :slight_smile:

Thanks. I made them in PowerPoint 2007, which makes graphics pretty easy. You can see the powerpoint file here:

Fantastic Kalid! I consider myself a pretty bright guy and work with numbers every day but e has always been an opaque subject to me. Ever since high school it’s been hanging out with its friend the logarithm with nothing to do but mock me! I think this article is going to help me think about e in new ways and I may even put it to use. I’m really glad I found your site!

Great Bob, I’m glad you liked it! Yeah, e and its pesky friends like natural logarithms were a thorn in my side for a while. It really bugs me when I use a concept without really knowing what it meant.

E got all this attention and I wanted to dig in and see what all the fuss was about :slight_smile:

I’ll be writing more on this topic as there are some really interesting ways of looking at “this constant, approximately equal to 2.71828…”

nice article, it’s so rare to find actual explanations when it comes to mathematical ideas, I was very happy to have found this one.

Thanks Jayson, the lack of intuitive explanations motivated me to start this site. I’m glad you are finding it useful :slight_smile:

I have a question. :slight_smile: you wrote that x=rate*time and that e^x=growth. using this information is it possible to find the smallest amount of time allowed, the gap between one time to the next. Does this give us the ability to give a value for that infinitesimally small value between each moment in time. Just a random ponderance. Nice guides specially enjoyed the vector calculus series. I was getting worried that i would have to take math as a second language. The words they use in calc. books are too big for me.

Hi Stephen, great question!

Yes, if you take the unit of time smaller and smaller, you will get the instantaneous rate of growth at a point. And the surprising thing is that this is e^x!

I want to write more about the calculus of it, but basically, when you have a certain amount of “stuff” (say, 10 units) then you are growing at 10 units per unit of time as well. Of course, once you grow just a little bit you have a new amount of “stuff” (10.1 units) and now you are growing at 10.1 per unit time.

It’s a bit mind-boggling, but it’s the way e works – the current instantaneous rate of growth is equal to the current amount. I’ll be writing more on this as I get a good, intuitive understanding of it :slight_smile:

Holy cow…somehow despite four semesters of calculus I forgot or failed to grasp that with any calculator I can do compound interest calculations as easily as circle areas. I’ve been dependent on financial calculators for twenty years. Now I’m going to read all your math-related posts. Keep writing!

That’s awesome Joe! I know what you mean, I had forgotten about being able to do compound interest as well – it’s funny how rarely we revisit old topics we’ve learned. E had always bothered me.

I hope you enjoy the other posts, I’ll keep cranking them out :slight_smile:

[…] After understanding the exponential function our next target is the natural logarithm. […]

Thanks for the wonderful explanation. However, at places, confusion seems to arise due to wrong use of terminology. e.g. instead of ‘e is the fundamental rate of change shared by all continually growing processes.’ it should be ‘e is the fundamental net growth in all continually growing processes (in a unit time that would account for 100% simple growth)’, if I have understood correctly. Elsewhere, you do say growth=e^rt, where r is the rate of growth. So e is the total growth & not rate of growth.

Hi Dr. Jani, that’s a great point – the current description of e is confusing. As you say, e^1 = e = the amount of continuous growth after 1 unit of time (assuming growing at a rate of 100% simple growth per unit time).

e^rt lets you compute the net growth for any rate and time. I’ll update the article to make this more clear, I appreciate the feedback!

Wow thank you so much for explaining e. The wikipedia article completely confused me about it

Thanks David. Yeah, it really bugs me when math topics are explained in a complicated way, I’m glad you liked it.

Kalid - thank you (and your contributors for their comments) for this site. I hope that your blog will be even more popular than Wiki and be the “goto” place for teachers and their students!

Thanks for the encouragement Sophie, you must have read my mind! I’m hoping this blog evolves into a place to think about new topics in a fresh, intuitive way, and share those “a-ha” moments from everyone.

Wikipedia is a good reference, but encyclopedias tend to focus on facts vs. understanding. There’s a place for both :slight_smile:

This is a great discussion. You should do the fundamental theorem of calculus, also.

Thanks Jobie, appreciate the comment. Yep, the Fundamental Theorem of Calculus is definitely on my list of upcoming topics :slight_smile: