Enter any positive number into the x value field and get back its log base 2 (log₂) — the binary logarithm. You'll also see the equivalent log base 10 and natural log (ln) values for comparison. Useful for computer science, information theory, and anything involving powers of 2.
Log Base 2 (log₂x)
--
Log Base 10 (log₁₀x)
--
Natural Log (ln x)
--
2^(log₂x) Verification
--
A logarithm answers the question: to what power must a base be raised to produce a given number? For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse operation of exponentiation.
You can calculate log₂(x) using the change-of-base formula: log₂(x) = log₁₀(x) / log₁₀(2), or equivalently ln(x) / ln(2). For example, log₂(32) = log(32) / log(2) = 5, because 2⁵ = 32.
log₂(256) = 8, because 2⁸ = 256. You can verify this by entering 256 into the calculator above.
Log base 2 is fundamental in computer science and information theory because computers operate in binary (base 2). It is used to measure information in bits, analyze algorithm complexity (e.g., binary search runs in O(log₂ n) time), and work with data structures like binary trees.
ln is the natural logarithm with base e (≈ 2.71828), while log₂ is the binary logarithm with base 2. They are related by: log₂(x) = ln(x) / ln(2). Natural log is common in mathematics and physics, while log base 2 is preferred in computer science.
Yes. If x is between 0 and 1, log₂(x) will be negative. For example, log₂(0.5) = -1 because 2⁻¹ = 0.5. The log of 0 is undefined, and the log of a negative number is not a real number.
log₂(1) = 0, because 2⁰ = 1. This holds true for any base: the logarithm of 1 is always 0, since any number raised to the power of 0 equals 1.
In computer science, log₂(n) frequently describes how many times you can halve n before reaching 1. Binary search, merge sort, and heap operations all have time complexities involving log₂(n), making it a critical concept for understanding algorithm efficiency.