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113 Math
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113.021.02 Logarithms - e

# 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:
$$e^{2 \times 5} = e^{10}$$