e = 2.718
e is the maximum possible result of continuously compounding 100% over one time period.
All of these things are equal:
$$e^{1/x} = 10$$
$$10^x = e$$
$$x = log_{10}(e)$$
$$x = [log_e(10)]^{-1}$$
$$log_e(10) \approx 2.3$$
Formula
Take yearly compounding, say 100% growth per period.
Given $1 you would have $2 after one period.
Split that one period into two periods and compound by 50% twice and you get
1. 1 + (1 * .5) = 1.5 for the first half and
2. 1.5 + (1.5 * .5) = 2.25
- Clearly better
100% annual rate, compounded four times per period:
1. 1 + (1 * .25) = 1.25
2. 1.25 + (1.25 * .25) = 1.5625
3. 1.5625 + (1.5625 * .25) = 1.953125
4. 1.953125 + (1.953125 * .25) = 2.44
It goes up more, but not by the same amount. The more you go up, the more it begins to converge around 2.71. Here's a table:
| n | $$(1 + \frac{1}{n})^n$$ |
|---|---|
| 1 | 2 |
| 2 | 2.25 |
| 4 | 2.441 |
| 12 | 2.613 |
| 365 | 2.7146 |
| 1,000 | 2.7169 |
| 10,000 | 2.7184 |
| 100,000 | 2.718268 |
| 1,000,000 | 2.7182804 |
Applicability for other growth rates
Often, e is given as the formula:
$$e^{rt}$$
If we wanted to denote 200% growth over a period of 5 years, it would be:
$$e^{2 \times 5} = e^{10}$$
Source:
- Logarithm Fundamentals https://www.youtube.com/watch?v=cEvgcoyZvB4
Tags:
- math