How To Think With Exponents And Logarithms

This website has helped me so much in so many different classes! I’m still confused as to why e is such a central number, though. I mean i get the logic behind taking the limit of (1+1/n)^n being e as your interval approaches continuity but why is does that number just happen to be 2.718… and what makes that number so much more special than any of the other infinite numbers on the numberline that it would be so important to so many different subjects and why does taking the derivative of this number to the x power always yield e^x? I asked my teachers these questions and the answer was more or less “because that’s the way natural growth works”. Is there some hidden underlying principle that I’m missing or just not getting?

Hi Mia, great question – check out the article on e for more details about how it works out to 2.718.

and this one on the various properties of e^x:

Now, on why it’s 2.718… and not some other number – that’s a tricky question. Why is the square root of two 1.414… and not some other number? Because 1.414… has the property that squaring it results in 2.

Similarly, 2.71828… has the property that it perfectly models continuous growth. We figure out the square root of 2 by thinking “Well, it must be greater than 1, but less than 2. Let’s try 1.5 – oh, that’s too big. How about 1.4? Close, but too small. How about 1.45? Too big. 1.41? Too small, but closer.” And so on – we iteratively walk towards the real value of the square root of 2.

For e, we figure it out by trying to model perfectly smooth growth, taking 100% interest without any gaps. If we take 100% interest in a single chunk, at the end of the year, we went from 1.0 to 2.0. But we can do better: take interest in 2 groups of 50%.

That gives us (1 + .5)^2 = 2.25. A bit better. We can take 3 groups of 33.3% interest (which gives us 2.37), or 4 groups of 25% interest, and keep going.

We hit a ceiling at 2.71828… that is, it’s our best approximation for what perfectly continuous growth looks like (just like 1.414… is our approximation of the square root of 2). We have an algorithm to give us a number with a certain property, and that’s what pops out. I can’t say why it’s 2.71828… and not 2.81828… except the latter does not match with what happens as you create growth that is smoother and smoother.

The idea of perfectly smooth growth also means e^x will be its own derivative. If we’re growing perfectly smoothly by 100%, our rate of change (d/dx) must equal our current value. (That’s what 100% smooth growth means: our growth trajectory is 100% of our current amount. 50% smooth growth means our trajectory is 50% of our current amount). In math terms,

d/dx current value = current value

and to get more specific,

d/dx e^x = e^x

Hope this helps.

Kalid–really great job explaining it all very intuitively. That is real teaching. I wish classroom teaching was on these lines. In my school days, I had to go find many different resources befor I could intuitively understand concepts. Way you explain it is still much better and giving me new ingights even now.

As to Mia’s question–why e=2.7… and not some other number,… here is my 2 cents, it is because we use decimal system for our numbers. and this natural number just falls at 2.7… in our decimal system, which was invented for our convenience (observer’s viewpoint)
Idealy if we were to re-invent number system from nature’s viewpoint, we should be using a "natural"number system–in which e woiuld fall at unity or “1”.

After reading this article there is one aha moment for me.complex lograthim can be explained in same way, if u take log of a rotation(final result: or observer view point ) the answer will give u the angle (mover’s point of view),plz comment on this ,many many thanks to share this article may god give u happiness and fun as you spread the fun in world through your precious insights

I start to understand the concept then get confused with calculations i.e.
14.4/9.9= .374 when I do it, I get 1.454 & 2.32/2.3= .0086 I get 1.0086 ???
Any help would be appreciated

First, Khalid you’re the best. Great article, excellence.

Some Insight, you have basically said this but I would like it to be said in pure calculus terms to understand laplace functions better:

Finding the continuous growth, by inputing “x” into e^x
-Where x=rate*time
–Integral of velocity with respect to time is distance (same as rate * time= growth where velocity is rate and dt (time) is a small portion of time and finally growth is the “distance”) (Better explanation, bottom paragraph)

Finding the cause of growth, AKA DIFFERENTIATION
ln(x) automatically gives you the rate of growth by inputting x. Same thing a derivative does.

Since e^x has a y output that has been integrated at every smallest portion of time (n tends to infinity);

  1. Input the x, which is a rate*time (already a Integrated value)
  2. Perform the operation of e^x (scales the integrated value(rate*time) by inputting the information of a continuous growth operation
    a) this means that depending on the size of your rate * time (if it is huge “rate”, or “time” (x depends on both) - the operation will give you a huge growth (distance, whichever way you want to think)
  3. Since it is a function that describes continuous rate of growth (a number that has been integrated infinite number of times), it will always be the derivative of itself.


@Brant: Make sure you’re taking the natural log, and doing ln(14.4/9.9) and not a regular (14.4/9.9).

@Magnus: Glad you’re enjoying it. Very interesting thought, e^x and ln(x) are analogous to integration/differentiation because they are both inverses and “undo” each other. And in general, e^x will make the quantity bigger (growing for some amount of time) and ln(x) will make the quantity smaller / finding the rate of change.

I think there are some good analogues here, it can be confusing because the integral/derivative of e^x is not ln(x), but the way they switch between cause and effect is similar. The derivative is the “cause” that creates a change [observed when you integrate the derivative to build up to a final effect].

It’s still too technical to understand with my non math brain. I was fried after the first paragraph.