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.
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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.