Crash Course on Logarithms 📘
By Naman Attri
🚀 Crash Course on Logarithms
Logarithms are one of the most useful mathematical tools — from calculating algorithmic complexity to dealing with exponential growth in real life.
In this quick crash course, you’ll understand what logarithms are, how they work, and how to apply them.
🧩 What is a Logarithm?
A logarithm answers the question:
“To what power must I raise this base to get that number?”
Formally:
\[ \log_b(a) = c \iff b^c = a \]Example
\[ \log_2(8) = 3 \text{ because } 2^3 = 8 \]🔢 Common Logarithm Bases
| Base | Name | Example |
|---|---|---|
| 10 | Common Logarithm | log₁₀(1000) = 3 |
| 2 | Binary Logarithm | log₂(32) = 5 |
| e (≈2.718) | Natural Logarithm (ln) | ln(e²) = 2 |
💡 Tip: In programming and algorithms,
logoften meanslog₂.
🧠 Why Do We Use Logarithms?
- Simplify exponential growth: Convert multiplication into addition.
e.g., \(\log(ab) = \log(a) + \log(b)\) - Measure complexity: Big-O like
O(log n)appears in binary search, tree height, etc. - Handle large numbers: Makes huge values manageable (e.g., sound intensity, earthquakes).
- Undo exponentials: Solve for unknown exponents easily.
⚙️ Logarithm Properties (Must-Know)
| Property | Formula | Example |
|---|---|---|
| Product | logₐ(xy) = logₐ(x) + logₐ(y) | log₂(8×4) = log₂(8) + log₂(4) = 3 + 2 = 5 |
| Quotient | logₐ(x/y) = logₐ(x) − logₐ(y) | log₁₀(100/10) = 2 − 1 = 1 |
| Power | logₐ(xᵖ) = p × logₐ(x) | log₂(8²) = 2 × 3 = 6 |
| Change of Base | logₐ(b) = log_c(b) / log_c(a) | log₂(8) = log₁₀(8) / log₁₀(2) = 0.903 / 0.301 = 3 |
⚡ Quick Practice
Try these:
\(\log_3(81) = ?\)
→ 81 = 3⁴ → Answer = 4\(\log_5(1/25) = ?\)
→ 1/25 = 5⁻² → Answer = -2\(\log_{10}(0.01) = ?\)
→ 0.01 = 10⁻² → Answer = -2
💻 Logarithms in Programming
| Language | Function | Example |
|---|---|---|
| Python | math.log(x, base) | math.log(8, 2) → 3.0 |
| JavaScript | Math.log(x)/Math.log(base) | Math.log(8)/Math.log(2) → 3 |
| Go | math.Log(x)/math.Log(base) | math.Log(8)/math.Log(2) → 3 |
| C++ | log(x)/log(base) | log(8)/log(2) → 3 |
📈 Real-World Applications
- Algorithms: Binary search, tree depth, sorting complexities.
- Finance: Compound interest & time to double money.
- Science: pH scale (logarithmic measure of acidity).
- Technology: Decibel scale for sound, Richter scale for earthquakes.
- AI & Data Science: Cross-entropy, log-likelihood in model evaluation.
🔍 Intuitive Analogy
Think of logarithms like counting how many times you multiply the base to reach a number.
Example:
| Step | Power of 2 | Value |
|---|---|---|
| 1 | 2¹ | 2 |
| 2 | 2² | 4 |
| 3 | 2³ | 8 |
| 4 | 2⁴ | 16 |
So, log₂(16) = 4, because you multiply 2 four times to reach 16.
🧮 Logarithmic Scales Simplify the Universe
| Phenomenon | Scale | Example |
|---|---|---|
| Sound | Decibel (dB) | Every +10 dB = 10× intensity |
| Earthquake | Richter | +1 magnitude = 10× stronger |
| Light | Luminosity scale | Logarithmic response of eyes |
| Information | Bits | log₂(n) represents choices among n outcomes |
🏁 Conclusion
Logarithms turn multiplication into addition, division into subtraction, and powers into multipliers.
They are everywhere — from coding algorithms to understanding the world’s natural scales.
So next time you see O(log n), just smile 😄 — you now know what that really means.
✅ Takeaway
A logarithm tells you how many times you multiply the base to get a number.
🧠 Written by Naman Attri
Built with ❤️ and curiosity for lifelong learners.