Question

Question #2ac6b

Answer 1

It's a way of measuring continuous growth.

Explanation:

The number `e` is very carefully constructed to measure continuous growth. You might've seen this limit for `e`:
`lim_(n->oo)(1+1/n)^n`

Basically what it is is if you multiply by a number infinitely close to `1` at every single instant.

What this means is that you can measure the amount of continuous growth after a certain time period `x` with the expression `e^x`.

You can also do the reverse and measure how much time it would take to get `x` growth using the natural log, log base `e`:
`ln(x)`.

If I wanted to figure out how long it would take me to grow to 10 times as much as I have now (if the growth is continuous, and the interest is a total of `100%` over a year), I'd take `ln(10)` to get roughly `2.3` years.

We can even take it one step further to think about things which have already happened. Say we wanted to figure out when I had half of what I have now (with the same interest and compounding conditions as before). I'd just take `ln(1/2)`, which is approximately equal to `-0.7`. So `0.7` years ago, I had half of what I have now.

You can also raise `e` to a negative exponent to figure out how much you had a certain amount of time ago. Suppose we wanted to figure out how much I had a year ago, we'd take `e^-1` to get around `0.4`. So a year ago, I had roughly `0.4` (or `2/5`) as much as I do now.

I think the reason `e` is useful to us is because a lot of things in nature don't increase at fixed intervals, rather they increase continually, which is why we need `e` to model how these processes work.