# 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