What does log base 2 actually tell you?

It counts doublings. log₂ x is the number of times you must double 1 to reach x, or equivalently the number of times you can halve x before reaching 1. Because those are counts, the answer is a whole number precisely when x is a power of two, and everything else lands between two whole numbers. That counting interpretation is why this particular base runs through computing, information theory and anything else built on repeated splitting in two.

Why is base 2 exact when other bases are not?

Because a double is already stored as a fraction times a power of two, so the exponent is sitting in the number waiting to be read. For a power of two the logarithm is a lookup rather than a computation, and it comes back with no rounding whatsoever. Checked across all 2,098 representable powers of two, the dedicated routine returned the exact integer on every one. No other base can do this: ln 8 and log₁₀ 8 are irrational, so there is no exact value to return in the first place.

Should I just compute it as ln x divided by ln 2?

No, and the reason is arithmetic rather than algebra. The change-of-base rule is perfectly true; the problem is that ln x and ln 2 are each rounded to fit a double, and dividing two rounded values does not restore what was lost. Measured in one engine, the division failed on 441 of the 2,098 representable powers of two and disagreed on 27.9 per cent of the first 200,000 integers; the precise counts vary between browsers because each ships its own log routine, and the panel on this page recounts them for yours. The differences are last-digit, but they cluster on the powers of two, which is the worst possible place for them.

How many bits do I need to store a number?

To store n distinct values you need ⌈log₂ n⌉ bits, because k bits hold exactly 2ᵏ different patterns. To write a particular integer n in binary you need ⌊log₂ n⌋ + 1 digits, which is a different question with often a different answer. For n = 8 the first gives 3 and the second gives 4: three bits hold eight values, while the number eight is 1000 and takes four digits. Both are shown on this page, side by side, because reaching for the wrong one is where buffer-sizing bugs come from.

What is the difference between floor, ceiling and bit length?

They answer three questions that sound alike. The floor is the position of the highest set bit, counting from zero. The ceiling is how many bits are needed to store that many distinct values. The bit length is how many binary digits the number itself occupies, which is always the floor plus one. For a power of two the floor and ceiling coincide and the bit length is one more; for everything else the ceiling equals the bit length and the floor is one less. Neither pattern is a general rule, which is why guessing between them fails.

Why does binary search take log base 2 steps?

Because each comparison halves what is left. Starting from n candidates, one step leaves n/2, two leave n/4, and after k steps you have n/2ᵏ. You are finished when that reaches 1, which happens at k = log₂ n. The worst case is the ceiling of that, since a partial step still costs a whole comparison. The same argument explains why a balanced binary tree over n items is about log₂ n deep, and why doubling your data adds only one more step rather than doubling the work.

What does it mean to measure information in bits?

Choosing one option from n equally likely ones carries log₂ n bits of information, because that is how many yes-or-no questions are needed to pin it down. One of 8 is 3 bits, one of 1024 is 10. When the options are not equally likely the same formula applies per outcome and is averaged, which is Shannon entropy — and it is why a fair coin flip is exactly one bit while a heavily biased one carries less. The unit is called a bit for precisely this reason: it is the information in one binary digit.

How do I work out log base 2 without a calculator?

For a power of two, count the halvings or the zeros: 1024 halves ten times, and it is a 1 followed by ten zeros in binary. For anything else, find the two powers of two either side to bracket it — 1000 sits between 512 and 1024, so its logarithm is between 9 and 10, and close to 10 since 1000 is nearly 1024. A useful shortcut is that log₂ 10 is about 3.32, so every factor of ten adds roughly 3.32: a million, being 10⁶, is near 19.9.

Where does log base 2 turn up in practice?

Wherever something splits or doubles. Sizing a buffer or hash table, choosing an integer width, the depth of a balanced tree, the cost of binary search or merge sort, the number of rounds in a knockout tournament, the octaves between two musical pitches, the stops on a camera, and the half-lives of anything decaying — all counts of doublings or halvings. It is also the natural unit of information, which is why file sizes, entropy and compression ratios are all quoted in bits.

Log Base 2 Calculator

Take the base-2 logarithm of any positive number, with the floor, the ceiling and the bit length kept apart — because those are three different questions and all three get called log2. Base 2 is the one logarithm that comes out exact. A double is stored as a fraction multiplied by a power of two, s...

log₂ x = the number of doublings from 1 to x

Equivalently, how many times x can be halved before it reaches 1. Where that count comes out whole, this logarithm is exact — and computing it as ln x ÷ ln 2 is what throws the exactness away.

THE NUMBER

LOG BASE 2 · EXACT

x is exactly 2 to the power 10, so this answer carries no rounding at all — the exponent was already stored in the number.

FLOOR

CEILING

BIT LENGTH

This is the case where a logarithm is exact rather than approximated. A double is stored as a fraction times a power of two, so for a power of two the answer is already sitting in the number and only has to be read out. That is unique to base 2 — ln 8 and log₁₀ 8 are both irrational, and no amount of care recovers a clean value that was never there. Note that 1024 needs 11 binary digits to write down even though its logarithm is 10; those are two different questions and the panel below keeps them apart.

Quantity	Value	What it answers
log₂ x	10	doublings from 1 to x
⌊log₂ x⌋	10	complete doublings, ignoring the remainder
⌈log₂ x⌉	10	bits to store x distinct values
bit length	11 (10000000000)	binary digits needed to write x
ln x	6.93147180559945	the same number scaled by ln 2
log₁₀ x	3.01029995663981	decimal digits, roughly
ln x / ln 2	10	agrees here, but not always

Floor, ceiling and bit length are three different questions. To store 1024 distinct values you need 10 bits. To write the number 1024 in binary you need 11 digits. And 10 is the position of its highest set bit. All three get called “log base 2” in conversation, and mixing them up is the most common source of off-by-one errors in code that allocates buffers or sizes hash tables.

THE HALVINGS

Count the rows and you have the logarithm. Each halving takes exactly one off it, which is why the answer for a power of two is a whole number and why the whole part is simply how many complete halvings fit.

Live calculation · updates as you type

Created with❤️byeaglecalculator.com

HOW TO USE

1
Type any positive number into x. The logarithm appears immediately, and if x is a power of two the panel says so — that answer carries no rounding at all, because the exponent was already stored in the number.
2
Read the floor, ceiling and bit length as three separate answers. To store x distinct values you need the ceiling; to write x in binary you need the bit length; the floor is the position of its highest set bit. They are not interchangeable.
3
Watch the middle panel if you were about to compute this as ln x ÷ ln 2. It marks in red every value where that division disagrees with the dedicated routine, computed live rather than quoted, and the disagreements sit on the powers of two.
4
Drag the slider to step through the halvings. Counting the rows gives the whole part of the logarithm directly, and the green row at the end — the leftover between 1 and 2 — is what supplies the fractional part.

REFERENCE FORMULAS

Rule	Formula	What it is for
Definition	log₂ x = y means 2ʸ = x	How many doublings from 1 reach x, or how many halvings of x reach 1.
Change of base	log₂ x = ln x / ln 2	True, but do not compute it this way: the division loses the exact integers.
Exact case	log₂(2ᵏ) = k	Exact for every representable k, because a double already stores its exponent.
Bits to write n	⌊log₂ n⌋ + 1	The number of binary digits. For 8 this is 4, since 8 is 1000.
Bits to store n values	⌈log₂ n⌉	Different question, often a different answer. For 8 distinct values, 3 bits.
Doubling	log₂(2x) = log₂ x + 1	Each doubling adds exactly one. This is why log scales in base 2 are readable.
Product	log₂(ab) = log₂ a + log₂ b	Multiplication becomes addition, as in any base.
Binary search	⌈log₂ n⌉ comparisons	The worst case for finding an item among n sorted ones by halving.
Information	log₂ n bits	The information in choosing one of n equally likely outcomes, measured in bits.
Inverse	2^(log₂ x) = x	Round-tripped on this page and reported in units in the last place.

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Last updated: August 3, 2026 · Computed with the dedicated base-2 routine, never by dividing two logs · checked against a 60-digit series