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.
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
ln(e) = 1
- the decimal logarithm, with base
log10(1000) = 100 = 10×10 = 10² = 2
- the binary logarithm, base
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.
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.
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.
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.
1a = 3 2 3log1 = np.log(a ** 3) 4log2 = 3 * np.log(a) 5 6print(log1) # 3.295836866004329 7print(log2) # 3.295836866004329
The logarithm allows for calculations of large numbers, it is the inverse of the exponential function.
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