An Intuitive Guide To Exponential Functions & e

Thanks Ganesh!

Hi to everyone, this is my first post, because I REALLY need help.

Everywhere i see know this advertisements about Forex thing, got attracted, started to read about it, but still really confused about What is forex .
All the explanations i find on the internet are very complicated. Too many smart words.

Can someone explain what is forex by his own words ?
Tired of websites.

Thanks.

[…] read and then comment (here) on this blog post:  http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ This entry was posted on Tuesday, February 21st, 2012 at 9:41 am and is filed under scribe […]

@Lin: Glad you liked it! Ack, that situation is all too common. Even if the teacher is excited and enthusiastic, they can still not explain it in a clear way. Unfortunately, we’re afraid to say the emperor has no clothes (“Nope, still not seeing it!”).

Please, tell me, As it applies to myself,

“Get you hungry for more: In the upcoming articles, I’ll dive into other properties of e.”

When will you “dive into”?

Thanks.
Javier.

@Jason: Awesome, glad you enjoyed it!

Hi Cody, the key to split interest into “n” parts and multiply it out. For earning interest every 6 months (i.e., 2 times a year) you get:

(1 + .5)^2 = (1 + .5) * (1 + .5) = 1 (original amount) + [.5 + .5] (the interest the original earned) + [.5 * .5] (interest the interest earned) = 1 + 1 + .25

You can “expand out” the multiplication for any n; for n = 5 you have (1 + 1/5)^5. However, it becomes cumbersome to track it like that when you can just compute the exponent. If you want to draw it out, divide the growth into 5 steps, and each step add a new interest element (Start with blue. Step 1: Blue earns Green; Step 2: Blue earns Green, Green earns Red; Step 3: Blue earns Green, Green earns Red, Red earns Orange; Step 4: Blue earns Green, Green earns Red, Red earns Orange, Orange earns Pink; Step 5: Blue earns Green, Green earns Red, Red earns Orange, Orange earns Pink; Pink earns Black).

Hi! you probably already read my comment about my essay paper! i was wondering if you could help me organize all my ideas to make it a little more simplistic! Thanks so much your website is great!

I’ve been looking for such an explanation for a long time now, much appriciated
imma go check some more articles out
many thanks for putting up this website

@Adrian: Awesome, glad it helped! e bothered me for a long, long time too :).

[…] off, take a break and learn about e’s evil twin, the natural logarithm. Original Article : http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/Pinaki Bhattacharyya Tags: exponential, logarithm, mathematics (function() { var po = […]

Amazing, thank you so much. I’ve been trying to get a better understanding of what it is that i’m learning rather than memorizing formulae, your detailed explanations are a real help. Kudos!

After reading your explanation, I understand (I hope!) that e is a constant, because all rates of growth are equal to the growth at 100%, but they have been squeezed or stretched, lengthened or shortened to accommodate the variations. (Is that right?) Your comparison to pi made it clear to me- all circles have the same curve, whether they are the edge of a dime or the edge of the sun. Thanks very much!

how would i find the approximate equation of a line that is similar to a decaying exponential function? i have a line that does decrease over time at a slowing rate, though instead of constant decay it sometimes repeats numbers as y approaches 0, and actually does reach 0 as there cant be half of the unit that im using.

hopefully im making some sense here… instead of constant decay, there is a chance of decay. so starting with one hundred, we lose alot, then some of those, some of the remaining until say, 2 are left. then those have a chance of going but dont for a few t units, then 1 left for a few times until it completely dissapears.

cheers!

Hi Moisés, great question. Your formulas have the rate as “-r” so we are shrinking. e^(-rt) is the formula for continuous, compound shrinking and (1-r)^t is the formula for compound but not continuous shrinking.

With continuous shrinking, we are shrinking at every possible instant. With non-continuous shrinking, the “shrinks” only happen at set intervals (every 3 months, for example). Check out this article for the difference:

http://betterexplained.com/articles/a-visual-guide-to-simple-compound-and-continuous-interest-rates/

Hi Coi, really glad it’s helping!

Thanks for the explanation!! I am a statistics major and I am embarrassed to report I do not know the explanation until today! Thanks!

@Martha: That’s pretty much it. e^x assumes 100% growth rate for time “x”. If you adjust the rate (50% growth, 200% growth) it’s like scalene e up and down (e^0.5x or e^2x), which is like changing the radius of a circle (to shrink it to a dime, or expand it to the sun). The core pattern is the same.

Hi Mikala, that’s awesome! I love it when people can share their knowledge with others :).

Khalid, Great article. But I am confusing myself. I thought i understood it but as I thought more and more about it I have confused myself.

Take following example -
If I have $1 that grows at a rate of 200% in 1 year with TWO time period (i.e. 100% per 1/2 year). by e ( rate * time) this will be e^2. But if I use your Mr Blue/Mr Green theory this gives me an answer of $4 ( if I have done it correct). I think I am missing something here. Please can you help?