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Log Base 2 Calculator

Enter a positive number to get log2(x), how many times you can halve it.

Calculate
A positive number. You can type 1/8, 2^10 or e.
Try:

Logarithm

log⁡2(1000)≈9.96578428466\log_{2}(1000) \approx 9.96578428466log2​(1000)≈9.96578428466

9.96578428466

Shown as a decimal to 12 significant digits (double precision).

Step by step

  1. 1Definition
    log⁡2(1000)=y\log_{2}(1000) = ylog2​(1000)=y
    ↓
    2y=10002^{y} = 10002y=1000

    The log asks: what power of the base gives the number?

  2. 2Change of base (natural logs)
    log⁡2(1000)=ln⁡(1000)ln⁡(2)\log_{2}(1000) = \frac{\ln(1000)}{\ln(2)}log2​(1000)=ln(2)ln(1000)​
    ↓
    6.907755278980.69314718056=9.96578428466\frac{6.90775527898}{0.69314718056} = 9.965784284660.693147180566.90775527898​=9.96578428466
  3. 3Product and power rules
    log⁡2(1000)\log_{2}(1000)log2​(1000)
    ↓
    3log⁡2(2)+3log⁡2(5)=3×1+3×2.321928094893\log_{2}(2) + 3\log_{2}(5) = 3\times1 + 3\times2.321928094893log2​(2)+3log2​(5)=3×1+3×2.32192809489

    log of a product is the sum of the logs.

The same number in other bases

Natural logln⁡x\ln xlnx6.90775527898
Common loglog⁡10x\log_{10} xlog10​x3
Binary loglog⁡2x\log_{2} xlog2​x9.96578428466
  • Definitionlog⁡bx=y  ⟺  by=x\log_b x = y \iff b^{y} = xlogb​x=y⟺by=x
  • Productlog⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b ylogb​(xy)=logb​x+logb​y
  • Quotientlog⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b ylogb​(yx​)=logb​x−logb​y
  • Powerlog⁡b(xn)=nlog⁡bx\log_b(x^{n}) = n\log_b xlogb​(xn)=nlogb​x
  • Change of baselog⁡bx=ln⁡xln⁡b=log⁡10xlog⁡10b\log_b x = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}logb​x=lnblnx​=log10​blog10​x​
  • Same baselog⁡bb=1,log⁡b1=0\log_b b = 1,\qquad \log_b 1 = 0logb​b=1,logb​1=0
  • Inverseblog⁡bx=x,log⁡b(bx)=xb^{\log_b x} = x,\qquad \log_b\left(b^{x}\right) = xblogb​x=x,logb​(bx)=x
  1. Tools
  2. Mathematics
  3. Logarithm Calculator
  4. Log Base 2 Calculator

About Log Base 2 Calculator

This log base 2 calculator finds log2(x), the power of 2 that gives x. It is the binary logarithm used throughout computing and information theory. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The page opens on log2(1000), about 9.96578428466. That means 2 to the power 9.97 is 1000, so it takes about 10 bits to count to 1000. Powers of two give exact answers: log2(1024) = 10 and log2(1/8) = -3.

It answers questions such as how many bits are needed to store N values (the ceiling of log2 N) and how many times a list can be halved in a binary search.

How to use Log Base 2 Calculator

  1. 1Choose a mode: log of any base, ln, log10, log2, antilog or solve b^x = y.
  2. 2Enter the base (not needed for ln, log10 and log2) and the number. You can type fractions like 1/8, powers like 2^10 or the constant e.
  3. 3Read the exact or decimal answer at the top.
  4. 4Open the steps for the definition, the change of base and, when the number is an integer, the product and power rules applied to its prime factors.
  5. 5For an exponential equation such as 1.05^x = 2, choose Solve b^x = y.

Features

  • Log of any base, with exact rational answers when the number is an exact power of the base.
  • Natural log, common log and binary log as one-click modes.
  • Antilogarithm: b raised to the power y, exact for integer exponents.
  • Solve b^x = y, the form of compound interest, growth and half-life questions.
  • Change-of-base steps using both ln and log10.
  • Product and power rules shown on the prime factors of an integer argument.
  • Related logs in bases e, 10 and 2, and a list of the logarithm rules.
  • Plain explanations of domain errors.

Binary logarithm

Compute it with natural or common logs.

log2(x) = ln(x) / ln(2) = log10(x) / log10(2)
log2(2^n) = n

Worked examples

  • log2 1000 → gives ≈ 9.96578428466

    Between log2(512) = 9 and log2(1024) = 10.

  • log2 1/8 → gives -3

    2^-3 = 1/8.

Privacy: how it works

  • Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • To check a log, raise the base to your answer. It should give back the number.
  • log of a number between 0 and 1 is negative, and log of 1 is 0 in every base.
  • For doubling time at a rate r per period, solve (1 + r)^x = 2 with the Solve mode.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Numbers are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

How many bits do I need for N values?

The smallest whole number of bits that is at least log2 N. For 1000 values that is 10 bits, because log2 1000 is about 9.97.

What is log2 of 1024?

Exactly 10, because 2^10 = 1024.

More Logarithm Calculator pages

  • Logarithm CalculatorPick the logarithm you need, enter the base and the number, and get the answer with the change-of-base working.
  • Natural Log CalculatorEnter a positive number to get ln(x), the logarithm to base e.
  • Log Base 10 CalculatorEnter a positive number to get log10(x), the common logarithm.
  • Antilog CalculatorEnter a base and an exponent to get the antilogarithm, b raised to y.
  • Change of Base CalculatorEnter a base and a number to see log_b(x) worked out with the change-of-base formula.
  • Solve Exponential EquationsEnter a base b and a target y to solve b^x = y for x.

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