An Intuitive Guide To Exponential Functions & e

This was really cool - thanks for taking the time to writ it up. I realy have a grasp of e now!

Awesome Dan, I’m happy it helped!

Very nice explanation. Even though I have a BS in Engineering, I still didn’t really know what e was. You brought back some memories and helped reinforce the ol’ e.

Your intuitive explanation reminded me of this article on impedance: http://sci-toys.com/attention/2006/04/impedance-matching.html

BTW, what is your profession and education?

Thanks!

Hi Jut, I’m glad you liked it. I find there’s many subjects we just plow through without really understanding – that feeling has always bothered me :).

I’ve got a BS in Computer Science and do programming/web development now, but enjoy a wide variety of subjects (there’s more in the about section).

I think I get this whole expotental thing a little more. Thanks! In class, they kind of throw this thing at you.

Thanks Alice, that’s great! Yeah, things become much easier when you have an intuitive understanding, I’m happy it was able to help.

Thanks man!!!

I have a BE in CSE ,still could not get what ‘e’ is.
passing exams and understanding the concepts is really totally different.Thanks again.
Please keep it up.

Hello, what about exponential decay when the rate is negative? With positive rates it is easy to understand but for some reason I can’t see it with negative rates

Hi Jayson, negative exponents can be tough. Positive exponents imply “growing” and negative exponents mean “shrinking”.

So, e^3 is “The amount of growth after 3 periods of time (assuming 100% growth rate, compounded).”

What’s e^-3? Well, it’s a growth rate of -100%, but what does that mean?

If you could shrink for 100% for one unit of time, you’d end up with zero. However, e is about continuous growth.

So, as you shrink a little bit, your rate of shrinking slows down. Instead of shrinking at 100%, maybe you’re shrinking at 90%.

As you shrink more, your rate may decrease to 80%. Then 70%. Eventually, your rate of shrinking is around 1% and it looks like you aren’t shrinking at all.

Thinking about it more, negative exponents are a bit strange :).

Another way is to consider negative exponents as negative time, rather than negative growth.

e^3 * e^-3 = 1. So, e^-3 can mean “What value do I start with, and grow for 3 units, to get my current value?”

e^-3 would be a really, really small number if it’s allowed to grow. When we look forward, we see growth. When we look backwards, we see decay.

After all, 100 dollars in 2007 might look like “decay” when you look back at it from 2010 (when you have 1000 dollars). In some cases, an amount really is shrinking as time goes forward (and looking back in time looks like growth).

A bit confusing, but hope this helps.

Can we look forward for more math explanation. The Articles you write can change the world (majority ) view on mathematics. We need such good teachers in our schools. Thanks

Thanks for the encouragement Taz – I have more posts in the works, but want to get them to high enough quality. I love helping people learn, and if I can help people enjoy “hated” subjects like math all the better :slight_smile:

Awesome explanation :slight_smile:
Something I wanted to know for a long time.
Please write more of the same kind of articles !

Well…! Finding good teachers is like finding a treasure.
I got to your explanation by… serendipity. :slight_smile:
Thank you very much!
Internet allows finding people who “just” do extraordinary pieces of work.
All the best!

Hi Nasir and Michael, thanks for the kind words! I have more articles in the works, so I hope you enjoy them as well.

I completely agree about the Internet in terms of bringing people together and discovering new things – I’m so happy I was born in this “modern” time period :).

The best explanation of e and ln and have ever seen, and I have looked before. Absolutely awesome.

Wow, thanks Dan, happy you liked it. I try to explain things the way I wish I were taught.

What about -e to the x power? My daughter’s pre-cal teacher gave this as homework and test question to graph. I believe the base should be positive, else you will get a hilly graph. Please explain

Hi Sophia, good question. Did the teacher mean -(e^x) or (-e)^x?

If it’s the first, it looks like the reverse of the e^x and isn’t too hard to graph.

If it’s the second (negative base), things get tricky. For whole numbers like (-e)^1, (-e)^2, (-e)^3 you can plot it out, and it will be hilly, jumping from negative to positive.

The tricky thing comes with intermediate values like (-e)^.5. This means the square root of -e, which is an imaginary number! Any non-integer exponent will have this problem.

So, the only real values you can plot are for the integers 1,2,3,4,5, etc. I’d ask the teacher to double-check for a typo, but otherwise you can only plot it out for the integer values. :slight_smile:

Thanks a bunch Kalid. You confirmed what I thought. I intend to ask the teacher about this again. My objective is for my daughter and the rest of the class to fully understand the concepts as they are introduced.

Thanks so much.

No problem, glad it was useful for you!