Compute the base-2 logarithm of any positive number, or work backwards to find the number from its logarithm. Enter a value for x to get log₂(x), or enter log₂(x) to get x — leave the other field blank and press Calculate.
The log base 2 calculator is a simple but powerful tool for working with binary logarithms. Give it a number and it returns the base-2 logarithm; give it a logarithm and it returns the original number. The base-2 logarithm — often written log₂(x) and also called the binary logarithm — answers a single, very practical question: to what power must 2 be raised to produce x? Because so much of computing, information theory, and mathematics is built on powers of two, this is one of the most useful logarithms you can know how to calculate.
Whether you are a student learning logarithms for the first time, a programmer sizing a data structure, or an engineer working with signals and information, this calculator gives you fast, accurate answers and shows the formula behind them. Below you will find a full explanation of what a base-2 logarithm is, how to calculate it by hand, worked examples, common real-world uses, and answers to frequently asked questions.
A logarithm is the inverse of exponentiation. The base-2 logarithm of x is the exponent to which the number 2 must be raised to give x. In symbols:
if 2^y = x, then log₂(x) = y
For example, 2 raised to the power 3 is 8 (2 × 2 × 2 = 8), so the base-2 logarithm of 8 is 3. We write this as log₂(8) = 3. In plain language: "2 to the power of 3 equals 8," and "the log base 2 of 8 is 3" are two ways of saying the same thing. The logarithm simply reads the exponent off the equation.
The number 2 here is called the base of the logarithm. Other common bases include base 10 (the common logarithm, log or log₁₀) and base e ≈ 2.71828 (the natural logarithm, ln). Base 2 is special because it counts in doublings: each time x doubles, its base-2 logarithm goes up by exactly 1.
Most calculators and programming languages do not have a base-2 logarithm button directly, but they always have a natural logarithm (ln) or a base-10 logarithm (log). To compute a base-2 logarithm from these, you use the change-of-base formula:
log₂(x) = ln(x) / ln(2) = log(x) / log(2)
Here ln(2) ≈ 0.693147 and log₁₀(2) ≈ 0.301030. So to find the base-2 logarithm of any number, take its natural logarithm and divide by 0.693147. This calculator uses exactly this method, which is why its results match any scientific calculator or spreadsheet to full precision.
For instance, to find log₂(10): ln(10) ≈ 2.302585, and 2.302585 ÷ 0.693147 ≈ 3.321928. So log₂(10) ≈ 3.32.
The calculator is bidirectional. If you already know the logarithm and want the original number, it simply raises 2 to that power, because exponentiation undoes the logarithm:
x = 2^(log₂(x))
So if log₂(x) = 5, then x = 2⁵ = 32. Enter 5 in the log₂(x) field, leave x blank, and the calculator returns 32. This is handy whenever you have a logarithmic quantity — such as a number of bits — and want to know the actual count it represents.
Base-2 logarithms of exact powers of two are whole numbers, which makes them easy to check:
Numbers that are not exact powers of two give non-integer results:
The default in this calculator is x = 8, which returns log₂(8) = 3. Press Calculate to see the result and the step-by-step formula, then replace 8 with your own number.
The base-2 logarithm has a beautifully intuitive meaning: it tells you how many times you must double 1 to reach x, or equivalently how many times you can halve x before reaching 1. Start at 1, double it to get 2, double again to 4, again to 8 — after three doublings you have reached 8, which is exactly why log₂(8) = 3.
This "counting doublings" interpretation is why base 2 appears everywhere that things split or combine in twos: binary numbers, which use only 0s and 1s; data storage measured in bits and bytes; algorithms that repeatedly halve a problem; and processes that grow by doubling. Whenever a quantity relates to powers of two, the base-2 logarithm is the natural way to measure it.
The binary logarithm is fundamental to computing. A single bit stores two possible values, two bits store four, three bits store eight, and in general n bits store 2ⁿ values. Turning that around, the number of bits needed to represent N distinct values is log₂(N), rounded up. So to store 1,000 different values you need log₂(1000) ≈ 9.97, rounded up to 10 bits. This calculation underlies everything from memory addressing to file formats.
In computer science, the efficiency of many algorithms is described using base-2 logarithms. A binary search through a sorted list of N items takes about log₂(N) steps, because each comparison halves the remaining search space. Searching a million items therefore takes only about 20 steps, since log₂(1,000,000) ≈ 19.93. Sorting algorithms such as merge sort and quicksort run in "N log N" time, where the log is base 2. This is why logarithmic performance is prized: it barely grows even as data explodes.
Claude Shannon built information theory on the base-2 logarithm. The information content of an event with probability p is −log₂(p) bits, and the entropy of a data source is measured in bits using base-2 logs. This is the theoretical foundation of data compression and digital communication: it tells us the minimum number of bits needed to encode a message.
Musical pitch is logarithmic, and octaves are doublings of frequency, so base-2 logarithms describe how many octaves separate two notes. The number of octaves between frequencies f₁ and f₂ is log₂(f₂/f₁). Because an octave is a factor-of-two change in frequency, base 2 is the perfect tool for measuring musical intervals in octaves.
In photography, each "stop" of exposure doubles or halves the light, so stops are base-2 logarithms of the light ratio. Biologists use base-2 logs to count cell-division generations, and the concept appears anywhere growth or decay happens by repeated doubling or halving.
Base-2 logarithms obey the same algebraic rules as all logarithms, which makes them easy to manipulate:
These identities let you break complicated expressions into simple sums and differences, which is exactly why logarithms were invented centuries ago to turn hard multiplication into easy addition.
You can only take the logarithm of a positive number. There is no real number y for which 2ⁿ equals zero or a negative number: raising 2 to any power always gives a positive result, growing without limit for large exponents and shrinking toward (but never reaching) zero for large negative exponents. As a result:
This calculator will let you know if you enter a non-positive value for x, since no real base-2 logarithm exists there.
All logarithms measure the same idea — the exponent needed to reach a number — but each base suits different problems:
Thanks to the change-of-base formula, you can convert between them at any time. For example, log₂(x) is always about 3.32 times log₁₀(x), because 1 / log₁₀(2) ≈ 3.32. Knowing this factor lets you estimate base-2 logs quickly from a base-10 result.
You can approximate base-2 logarithms in your head with a couple of tricks. First, locate your number between two powers of two. Since 2⁶ = 64 and 2⁷ = 128, you know log₂(100) is between 6 and 7 — and because 100 is roughly two-thirds of the way from 64 to 128 on a ratio scale, the answer (6.64) sits in that range. Second, remember that multiplying a number by 2 adds exactly 1 to its base-2 logarithm, and multiplying by 10 adds about 3.32. These shortcuts are great for sanity-checking the calculator's output.
It is the power to which 2 must be raised to produce that number. For example, log₂(8) = 3 because 2³ = 8. It effectively counts how many times you must double 1 to reach the number.
Use the change-of-base formula: log₂(x) = ln(x) / ln(2), or equivalently log(x) / log(2). Take the natural or common logarithm of your number and divide by ln(2) ≈ 0.693147 (or log₁₀(2) ≈ 0.301030).
Approximately 3.321928. Since 10 is not a power of two, the result is not a whole number. It lies between log₂(8) = 3 and log₂(16) = 4, as expected.
No, not within the real numbers. The base-2 logarithm is only defined for positive values, because 2 raised to any real power is always positive. As x approaches 0, log₂(x) tends toward negative infinity.
Computers store data in bits, and n bits represent 2ⁿ values, so the number of bits needed for N values is log₂(N). It also describes the efficiency of algorithms like binary search and appears throughout information theory.
They use different bases: log₂ uses base 2, while ln (the natural log) uses base e ≈ 2.71828. They are related by log₂(x) = ln(x) / ln(2), so a natural log can always be converted to a base-2 log by dividing by 0.693147.
This Log Base 2 Calculator is provided for general educational and informational purposes. Results are computed using the change-of-base formula and standard floating-point arithmetic; for exact powers of two the answers are whole numbers, while other inputs are shown to high precision. Always verify critical calculations independently where accuracy is essential.