Log Base 2 Calculator
Calculate the binary logarithm (log base 2) of any positive number. Log base 2 answers the question: "2 raised to what power equals this number?" It is fundamental in computer science, information theory, binary systems, and algorithm complexity analysis.
Formula
The binary logarithm can be computed using the change of base formula. Since ln(2) ≈ 0.693147, you can divide the natural log of any number by 0.693147 to get its log base 2.
Log Base 2 Reference Table
| x | log₂(x) | Meaning |
|---|---|---|
| 1 | 0 | 2^0 = 1 |
| 2 | 1 | 2^1 = 2 |
| 4 | 2 | 2^2 = 4 |
| 8 | 3 | 2^3 = 8 |
| 16 | 4 | 2^4 = 16 |
| 32 | 5 | 2^5 = 32 |
| 64 | 6 | 2^6 = 64 |
| 128 | 7 | 2^7 = 128 |
| 256 | 8 | 2^8 = 256 |
| 512 | 9 | 2^9 = 512 |
| 1,024 | 10 | 2^10 = 1,024 |
| 2,048 | 11 | 2^11 = 2,048 |
| 4,096 | 12 | 2^12 = 4,096 |
| 8,192 | 13 | 2^13 = 8,192 |
| 65,536 | 16 | 2^16 = 65,536 |
| 1,048,576 | 20 | 2^20 = 1,048,576 |
Frequently Asked Questions
What is log base 2 used for?
Log base 2 is used in computer science (binary search has O(log₂ n) complexity), information theory (bits of information), audio engineering (octaves), and any system based on powers of 2.
What is log₂(1)?
log₂(1) = 0, because 2⁰ = 1. The logarithm of 1 in any base is always 0.
What is log₂(0)?
log₂(0) is undefined. The logarithm of zero does not exist because no power of 2 can equal zero.
How is log₂ related to bits?
The number of bits needed to represent N distinct values is ⌈log₂(N)⌉. For example, 256 values need log₂(256) = 8 bits (1 byte).
Log Base 2 vs Log Base 10 vs Natural Log
The three most common logarithm bases serve different purposes. Log base 2 (binary logarithm, lb) is fundamental in computer science and information theory — it tells you how many bits are needed to represent a value. Log base 10 (common logarithm, log) is used in engineering, decibel calculations, pH chemistry, and the Richter scale. Natural log (ln, base e ≈ 2.718) appears in calculus, continuous growth/decay, and physics. They are all related by the change of base formula: log_b(x) = ln(x) / ln(b).
Applications in Computer Science
- Algorithm complexity: Binary search runs in O(log₂ n) time — searching 1 million items takes only 20 comparisons
- Data structures: Balanced binary trees (AVL, Red-Black) have height log₂(n), giving O(log n) lookup
- Information theory: Shannon entropy measures information in bits using log₂ — coined by Claude Shannon in 1948
- Networking: IP subnetting uses powers of 2 — a /24 subnet has 2⁸ = 256 addresses
- Audio: Each octave doubles frequency — the number of octaves between two frequencies is log₂(f₂/f₁)
- Compression: Huffman coding efficiency is measured against the entropy (log₂-based) of the source
Powers of 2 in Computing
Understanding log base 2 is inseparable from understanding powers of 2 in computing. Memory sizes (RAM, storage), bus widths, color depths, and addressing all use powers of 2. A 32-bit system can address 2³² = 4,294,967,296 bytes (4 GB). A 64-bit system can theoretically address 2⁶⁴ = 18.4 exabytes. The log₂ of these values tells you the bit width needed: log₂(4 GB) = 32 bits.