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Log Base 2 Calculator | Compute Base-2 Logarithms Instantly

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Compute the base-2 logarithm of any number with the change-of-base formula shown, plus the exact result for powers of two. Essential for information theory, algorithm analysis, binary math and computer science coursework.

Last updated: June 2026 · Free · No sign-up required

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Results

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Quick reference

xlog₂(x)
10
21
83
102410
0.5−1

info Log Base 2 Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

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info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

log₂(x) signMeaning
x = 1log₂ = 0
x > 1Positive — x is more than 1 doubling
0 < x < 1Negative result
x is a power of 2Whole-number result
x ≤ 0Undefined

lightbulb Worked example: bits needed for 50,000 IDs

A database needs unique binary IDs for 50,000 products. How many bits wide must the ID field be? Compute log₂(50000) = ln(50000)/ln(2) ≈ 10.8198/0.6931 ≈ 15.61. Since 2^15 = 32,768 is too small and 2^16 = 65,536 covers the range, the field needs 16 bits.

Result: log₂(50,000) ≈ 15.61 — round up to 16 bits.

What this means: Fifteen bits can only address 32,768 items, which falls short of 50,000, while 16 bits cover 65,536. Rounding the logarithm up gives the required address width; the fractional part (0.61) shows how much headroom the 16-bit space leaves before it overflows.

Formula & methodology

Formula: log₂(x) = ln(x) / ln(2) = log₁₀(x) / log₁₀(2)

How log₂ is computed

log₂(x) answers “2 raised to what power equals x?”. Because most math libraries only provide natural or base-10 logarithms, the result uses the change-of-base identity:

log₂(x) = ln(x) / ln(2) = log₅₀(x) / log₅₀(2)

  • Take the natural log of x and divide by ln(2) ≈ 0.6931.
  • Equivalently, take log₅₀(x) and divide by log₅₀(2) ≈ 0.30103.
  • Exact powers of two return whole numbers: log₂(1024) = 10.

Reading the result

Non-integer results are normal: log₂(50,000) ≈ 15.61 means 2^15.61 ≈ 50,000. In computing, rounding that value up tells you how many bits an index needs — 16 bits here, since 2^15 = 32,768 is too small.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do you calculate log base 2?
Use the change-of-base formula: divide the natural log of the number by the natural log of 2, or the base-10 log of the number by the base-10 log of 2. For example, log₂(40) = ln(40) ÷ ln(2) ≈ 3.689 ÷ 0.693 ≈ 5.32. Any calculator with a log or ln button can do this, and the tool applies it automatically for any input.
What is log base 2 used for?
The Log Base 2 Calculator is a free online tool that log base 2 calculator is a free online calculator that helps you work out log base 2 quickly and accurately, right in your browser. You enter a few values, and it returns an accurate result together with a step-by-step explanation, so it works as both a calculator and a quick reference for the underlying concept.
What is log base 2 of 0?
The Log Base 2 Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Log Base 2 Calculator is a free online calculator that helps you work out log base 2 quickly and accurately, right in your browser. There is no limit on usage, and it works on any device.
Why is log₂ of a power of 2 a whole number?
Because log₂(x) asks 'how many times must 2 be multiplied by itself to reach x?' — and a power of 2 like 8 is exactly 2×2×2, so log₂(8) = 3 by definition. The logarithm counts the exponent, which is an integer exactly when x is a power of 2. Non-powers fall between integers: log₂(10) ≈ 3.32.
What is the relationship between log₂ and bits?
The number of bits needed to represent n distinct values is ⌈log₂(n)⌉ — each bit doubles the possibilities, so the bit count is the base-2 logarithm rounded up. An 8-bit byte represents 2⁸ = 256 values, and log₂(256) = 8. Information theory measures content in bits for the same reason: each bit is one binary decision.
How many bits do I need to store a given number of values?
Take log₂ of the count and round up. For 50,000 distinct IDs, log₂(50000) = ln(50000)/ln(2) ≈ 15.61, and 2^15 = 32,768 is too small while 2^16 = 65,536 covers it — so you need 16 bits. This is the standard address-space calculation in memory, storage, and networking design.
Why does log₂ of a number between 1 and 2 give a fraction?
Because log₂(x) answers ‘2 to what power gives x’, and fractional powers are valid exponents. log₂(1.5) ≈ 0.585 means 2^0.585 ≈ 1.5. Whole-number results happen only for exact powers of two like 8, 16, or 1024; everything in between lands on a fractional exponent.
How is log₂ used in algorithm analysis?
Binary search halves the problem each step, so searching n sorted items takes about log₂(n) comparisons. For a million records, log₂(1,000,000) ≈ 19.93 — roughly 20 comparisons instead of a million. That logarithmic growth is exactly why binary search and balanced trees scale so well.
What is the change-of-base formula behind the results?
log₂(x) = ln(x)/ln(2) = log₅₀(x)/log₅₀(2), using ln(2) ≈ 0.6931 or log₅₀(2) ≈ 0.30103. Most hardware and spreadsheet functions only compute natural or base-10 logs, so dividing by the log of 2 converts them exactly. The page shows both forms so you can reproduce the calculation anywhere.

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