In [1]:

```
import math
import numpy as np
```

What is **e**? It is simply a number (known as Euler's number):

In [2]:

```
math.e
```

Out[2]:

**e** is a significant number, because it is the base rate of growth shared by all continually growing processes.

For example, if I have **10 dollars**, and it grows 100% in 1 year (compounding continuously), I end up with **10*e^1 dollars**:

In [3]:

```
# 100% growth for 1 year
10 * np.exp(1)
```

Out[3]:

In [4]:

```
# 100% growth for 2 years
10 * np.exp(2)
```

Out[4]:

Side note: When e is raised to a power, it is known as **the exponential function**. Technically, any number can be the base, and it would still be known as **an exponential function** (such as 2^5). But in our context, the base of the exponential function is assumed to be e.

Anyway, what if I only have 20% growth instead of 100% growth?

In [5]:

```
# 20% growth for 1 year
10 * np.exp(0.20)
```

Out[5]:

In [6]:

```
# 20% growth for 2 years
10 * np.exp(0.20 * 2)
```

Out[6]:

What is the **(natural) logarithm**? It gives you the time needed to reach a certain level of growth. For example, if I want growth by a factor of 2.718, it will take me 1 unit of time (assuming a 100% growth rate):

In [7]:

```
# time needed to grow 1 unit to 2.718 units
np.log(2.718)
```

Out[7]:

If I want growth by a factor of 7.389, it will take me 2 units of time:

In [8]:

```
# time needed to grow 1 unit to 7.389 units
np.log(7.389)
```

Out[8]:

If I want growth by a factor of 1, it will take me 0 units of time:

In [9]:

```
# time needed to grow 1 unit to 1 unit
np.log(1)
```

Out[9]:

If I want growth by a factor of 0.5, it will take me -0.693 units of time (which is like looking back in time):

In [10]:

```
# time needed to grow 1 unit to 0.5 units
np.log(0.5)
```

Out[10]:

As you can see, the exponential function and the natural logarithm are **inverses** of one another:

In [11]:

```
np.log(np.exp(5))
```

Out[11]:

In [12]:

```
np.exp(np.log(5))
```

Out[12]: