Last update: February 11th, 2022

The logarithm function is used to calculate large numbers.

To calculate a logarithm, we use a base. Therefore, to understand the principle, you have to refer to the powers.

For example 100 = 10×10 = 10². The 10 is multiplied two times. The 10 is the base, and the logarithm value is equal to 2: log10(1000) = 2.

The logarithm is only calculated with positive numbers and greater than 1.  ### The 3 types of logarithms

There are 3 logarithms:

• the neperian logarithm (or natural logarithm), noted ln, with base e -> ln(e) = 1
• the decimal logarithm, with base 10 -> log10(1000) = 100 = 10×10 = 10² = 2
• the binary logarithm, base 2 -> log2(256) = 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 = 8 (steps)

The steps (the result) correspond to the number of divisions needed with the base before arriving at number one. As a reminder, you can't divide by zero. So we are always sure to have a positive number greater than 1—an interesting calculation for Machine Learning, for example.

#### Amusement Park II

Here the speed is illustrated with this roller coaster. In the first comic strip, the roller coaster shows fast, then progressively slows down to arrive at a cruising speed in the last comic strip, which corresponds to the arrival.

#### Code python
1x = np.linspace(-3,3,100)
2
3ln = np.log(x)
4decimal = np.log10(x)
5binary = np.log2(x)
6
7plt.plot(x, ln, label='$natural$')
8plt.plot(x, decimal, label='$decimal$')
9plt.plot(x, binary, label='$binary$')
10
11plt.title('Logarithms',fontsize=12)
12plt.savefig('logarithms.png', bbox_inches='tight')
13
14plt.xlabel('x')
15plt.ylabel('y')
16plt.grid()
17plt.legend()
18plt.show()

### Logarithm of a product

The logarithm of a product is the sum of the logarithms of its factors.

#### Code python
1a = 3
2b = 4
3
4log1 = np.log(a*b) # 2.48491
5log2 = np.log(a) + np.log(b) # 2.48491

### Logarithm of a quotient

The logarithm of a quotient is the difference of the logarithms of its two terms.

#### Code python
1a = 3
2b = 4
3
4log1 = np.log(a / b) # −0.287682
5log2 = res3 = np.log(a) - np.log(b) # −0.287682

### Logarithm of a power

The logarithm of a power is the product of the exponent and the logarithm of the base.

#### Code python
1a = 3
2
3log1 = np.log(a ** 3)
4log2 = 3 * np.log(a)
5
6print(log1) # 3.295836866004329
7print(log2) # 3.295836866004329

#### Formula  The logarithm allows for calculations of large numbers, it is the inverse of the exponential function.