— 4 Feb 2025

Logarithms: The Basics Explained Simply

A logarithm is the opposite of exponentiation. It answers the question:

👉 "What power should we raise a base to get a number?"

But wait… what exactly is a "power"? 🤔

The power of a number (also called an exponent) tells us how many times to multiply a number by itself.

For example, the expression:

2³  # 2 * 2 * 2 = 8

…means multiplying 2 by itself three times, resulting in 8. 🚀

Now that you understand exponents, logarithms (log) are just their reverse operation.

For example, the logarithm:

log₂(8) = ?

…is asking "What power of 2 gives 8?" 🤔

Since we already know:

2³ = 8

…the answer is:

log₂(8) = 3

Let’s say we multiply 2 by itself 9 times:

2 ** 9  # 512

Now, what if we only have 512 and we want to reverse this process?

We need to find how many times we multiplied 2 to get 512. That’s where logs come in!

import math

math.log(512, 2)  # 9

✨ This tells us that 2 was raised to the power of 9 to get 512.

NumPy provides a natural logarithm (log base e) function:

import numpy as np

x = np.array([1, 10, 100, 1000])
np.log(x)  # Natural log (ln) with base 'e'

If you need a logarithm with a different base, use:

np.log2(x)  # Log base 2
np.log10(x)  # Log base 10

In Deep Learning, we often use log() in calculations like log-likelihood and cross-entropy loss.

Here’s how to compute logs in PyTorch:

import torch

x = torch.tensor([1.0, 10.0, 100.0])
torch.log(x)  # Natural log (ln)

If you need log base 2 or 10, you can manually convert it:

torch.log2(x)   # Log base 2
torch.log10(x)  # Log base 10

In short, logs tell us the exponent we need to reach a certain number.

Think of it like this:

Exponentiation asks: "What’s the result when I multiply a base by itself?"
Logarithms ask: "How many times did I multiply the base to get this result?"

🔹 Logs are everywhere in Deep Learning, from loss functions to probabilities and gradient descent.

Hope this helped! 🚀 Let me know if you want more examples! 😃