Log Base 2 Calculator

What is log base 2?

The binary logarithm (log₂) answers the question: "2 raised to what power equals x?"

$$\log_2(x) = y \iff 2^y = x$$

Formula

Using the change of base formula:

$$\log_2(x) = \frac{\ln(x)}{\ln(2)} = \frac{\log_{10}(x)}{\log_{10}(2)}$$

Common values

x	log₂(x)
1	0
2	1
4	2
8	3
16	4
32	5
64	6
128	7
256	8
512	9
1024	10

Applications in computer science

Bits needed to represent n values

$$\text{bits} = \lceil \log_2(n) \rceil$$

For example, to represent 100 different values, you need ⌈log₂(100)⌉ = 7 bits.

Binary search complexity

Binary search has O(log₂ n) time complexity, meaning each step halves the search space.

Tree depth

A balanced binary tree with n nodes has height ≈ log₂(n).

Information theory

Log base 2 is fundamental to information theory. The entropy (information content) is measured in bits:

$$H = -\sum p_i \log_2(p_i)$$

Properties

$$\begin{aligned} \log_2(1) &= 0 \\ \log_2(2) &= 1 \\ \log_2(xy) &= \log_2(x) + \log_2(y) \\ \log_2(x/y) &= \log_2(x) - \log_2(y) \\ \log_2(x^n) &= n \cdot \log_2(x) \end{aligned}$$