Calculate logarithms, powers, and analyze binary data.
Calculates log₂(x).
log₂(x) = ln(x) / ln(2)ln(8) ≈ 2.079442 and ln(2) ≈ 0.6931472.079442 / 0.693147 ≈ 38 = 2^3, the exact value is 3.floor = 3, ceil = 3log₂(8) ≈ 3
Live Graph: y = log₂(x)
Paste one number per line. We compute log₂, floor/ceil, power-of-two and bit length.
| x | log₂(x) | floor | ceil | power-of-two | bit length |
|---|---|---|---|---|---|
| 8 | 3 | 3 | 3 | YES | 4 |
| 10 | 3.321928 | 3 | 4 | no | 4 |
| 16 | 4 | 4 | 4 | YES | 5 |
| 1024 | 10 | 10 | 10 | YES | 11 |
Master the binary logarithm. Understand the math, the applications, and the implementation details that drive modern computing.
A log base 2 calculator tells you how many times the number 2 must be multiplied by itself to reach a given value. For any positive number x, log base 2 of x returns the exponent y such that 2 raised to the power y equals x.
Log base 2 answers a simple question: “How many times must we multiply 2 by itself to reach a particular number?” For example, log base 2 of 8 equals 3 because 2 multiplied by itself three times equals 8. Similarly, log base 2 of 32 equals 5.
This follows the familiar sequence of powers of two: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16.... When you calculate log base 2 of 32, you are identifying the exponent that produces the number 32.
Log base 2 is also known as the binary logarithm because it describes how values grow or shrink in a system that operates on two states: 0 and 1. Modern computers, memory devices, and communication systems rely on binary logic, so log base 2 naturally appears whenever you measure capacity, efficiency, or complexity.
In data storage, the number of bits required to represent a range of values equals the ceiling of log base 2 of that value. For example, 256 distinct values require exactly 8 bits. One hundred possible values require about 6.64 bits, which rounds up to 7 bits.
In algorithms, expressions that involve log base 2 usually describe processes that repeatedly divide a problem into halves. Binary search, balanced tree traversal and many index structures have running times that grow in proportion to log base 2 of the input size rather than the input size itself.
Information theory measures uncertainty and compression efficiency using bits. Because each bit represents two possible states, log base 2 provides the foundation for entropy formulas, coding schemes, and compression methods.
If a device or library does not provide a direct log2 function, you can compute log base 2 using natural logarithms. The standard identity is: log2(x) = ln(x) divided by ln(2). This is the approach used internally by many calculators and programming environments.
Example: ln(50) is approximately 3.912 and ln(2) is approximately 0.693. Dividing these values gives log2(50) ≈ 5.64.
In Excel you can use the formula LOG(number, 2) or LN(number)/LN(2). In Google Sheets, the same formulas apply. In Python you can call math.log2(x). In JavaScript you can call Math.log2(x). In C, C++ or Java you can divide log(x) by log(2). Although the syntax changes, the underlying idea remains the same.
When x is between 0 and 1, log base 2 returns a negative value. This happens because the exponent needed to reach a fraction is negative. For example, 2 to the power minus 1 equals one half, 2 to the power minus 2 equals one quarter, and 2 to the power minus 3 equals one eighth. As a result, log2(1/2) = −1, log2(1/4) = −2 and log2(1/8) = −3.
Log base 2 also helps when analysing large values. It compresses wide numeric ranges into small exponents. For example, 1024 corresponds to 10, about one million corresponds to 20, and about one billion corresponds to 30. This view is useful when estimating storage requirements and algorithmic complexity.
| Value | Log2 | Power |
|---|---|---|
| 1 | 0 | 2⁰ |
| 2 | 1 | 2¹ |
| 4 | 2 | 2² |
| 8 | 3 | 2³ |
| 16 | 4 | 2⁴ |
| 32 | 5 | 2⁵ |
| 64 | 6 | 2⁶ |
| 128 | 7 | 2⁷ |
| 256 | 8 | 2⁸ |
| 512 | 9 | 2⁹ |
| 1024 | 10 | 2¹⁰ |
No. Only perfect powers of two produce whole numbers. All other values produce decimal results. For example, log2(8) is exactly 3, but log2(50) is approximately 5.64.
No. Real logarithms are defined only for positive numbers. There is no real exponent y such that 2 raised to the power y equals zero or a negative value. The calculator shows a clear error message if you enter x less than or equal to zero.
Digital systems use binary representation. Each bit has two possible states, so powers of two and log base 2 describe storage capacity, addressing ranges, branching depth and many algorithmic processes in a natural way.
All three are logarithms, but they use different bases. Log10 works with base 10 and is common in decimal and scientific notation. The natural log uses the constant e and is used in continuous-growth and calculus problems. Log2 uses base 2 and is the standard choice whenever data or processes are organised in binary form.
A log base 2 calculator is more than a simple input–output widget. When it is supported by clear explanations and real-world context, it becomes a practical learning and problem-solving resource. It allows students to connect theory with application, developers to reason about complexity and storage, and professionals to understand how binary structures behave at scale. By presenting log2(x) with precise definitions, intuitive examples, platform-specific guidance and relevant applications, you provide users with both accurate results and genuine understanding. That combination turns a basic calculator page into a professional, authoritative reference on log base 2.