Log Base 2 Calculator
Calculate binary logarithm (log₂) for any positive number
Must be a positive number
How to Use This Calculator
Enter Number
Input the positive number (x) for which you want to find the binary logarithm. The number must be greater than 0.
Calculate
Click "Calculate log₂" to compute the binary logarithm. The result shows what power of 2 equals your input number.
Verify Result
Check the verification: 2 raised to the power of the result should equal your input number.
Formula
log₂(x) = ln(x) / ln(2) = log₁₀(x) / log₁₀(2)
Where:
- x = positive number
- log₂(x) = binary logarithm of x
- If 2y = x, then log₂(x) = y
Example 1: log₂(8)
2y = 8
2³ = 8
Therefore: log₂(8) = 3
Example 2: log₂(16)
2y = 16
2⁴ = 16
Therefore: log₂(16) = 4
Example 3: log₂(1024)
2y = 1024
2¹⁰ = 1024
Therefore: log₂(1024) = 10
About Log Base 2 Calculator
The Log Base 2 Calculator (also called Binary Logarithm Calculator) computes the binary logarithm of a number. Binary logarithm answers the question: "What power of 2 equals this number?" This is extremely useful in computer science, where binary (base-2) systems are fundamental, and in algorithm analysis where log₂ appears in time complexity calculations.
When to Use This Calculator
- Computer Science: Analyze algorithm time complexity (Big O notation with log₂)
- Binary Systems: Work with binary numbers, bits, and bytes
- Information Theory: Calculate information entropy and bit requirements
- Data Structures: Analyze binary search trees, heaps, and sorting algorithms
- Mathematics: Solve problems involving powers of 2
Why Use Our Calculator?
- ✅ Instant Results: Get accurate binary logarithm calculations immediately
- ✅ Computer Science Focus: Perfect for algorithm analysis and binary calculations
- ✅ Verification: Shows verification that 2^result equals your input
- ✅ Educational: Learn about binary logarithms and their applications
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: High precision calculations
Common Applications
Binary Search: In a sorted array of n elements, binary search takes at most log₂(n) comparisons. For 1,024 elements: log₂(1024) = 10 comparisons maximum.
Binary Tree Height: A complete binary tree with n nodes has height log₂(n+1) - 1. For 15 nodes: height = log₂(16) - 1 = 4 - 1 = 3 levels.
Information Theory: To encode n distinct values, you need at least log₂(n) bits. For 256 values: log₂(256) = 8 bits (1 byte).
Tips for Best Results
- The input number must be positive
- Power-of-2 values (2, 4, 8, 16, 32, 64, etc.) give integer results
- log₂(1) = 0, log₂(2) = 1, log₂(4) = 2, log₂(8) = 3, etc.
- Useful mental shortcut: log₂(n) ≈ number of times you can divide n by 2 until you reach 1
- For approximate calculations, log₂(x) ≈ 3.32 × log₁₀(x)
Frequently Asked Questions
What is log base 2 used for?
Log base 2 (binary logarithm) is used extensively in computer science for algorithm analysis (time complexity), binary systems, information theory, and data structures like binary search trees.
How do I convert log base 2 to natural log or log base 10?
Use the change of base formula: log₂(x) = ln(x) / ln(2) = log₁₀(x) / log₁₀(2). Alternatively, log₂(x) ≈ 3.32 × log₁₀(x).
Why is log base 2 important in computer science?
Because computers use binary (base-2) systems, many algorithms naturally involve log₂. Binary search, binary trees, divide-and-conquer algorithms, and sorting all have log₂ in their complexity analysis.
What's the relationship between log₂ and powers of 2?
If 2^n = x, then log₂(x) = n. For example, 2⁸ = 256, so log₂(256) = 8. Powers of 2 always give integer log₂ values.
How do I calculate log₂ mentally for small numbers?
Count how many times you need to divide the number by 2 to get to 1. For 16: 16 → 8 → 4 → 2 → 1 (4 divisions), so log₂(16) = 4.