Math calculator
Logarithms in any base, with the power they are really asking about.
What power gives this?
Base 10, 2, or the letter e for the natural log. Any positive base but 1.
log₂(1024)
10
Because 2¹⁰ = 1024.
The answer
10
exact — the power was checked
As a power
2¹⁰ = 1024
the same statement, the other way round
Base
2
written log₂
Sign
positive
the number is above 1
Change of base
A calculator with only ln and log₁₀ still does every base. Divide one logarithm by the other and the base cancels — which is why both lines above give the same answer.
The laws, on your numbers
| log(ab) = log a + log b | log₂(1024) + log₂(2) = log₂(2048) |
|---|---|
| log(a/b) = log a − log b | log₂(1024) − log₂(2) = log₂(512) |
| log(aⁿ) = n log a | log₂(1048576) = 2 × 10 = 20 |
| log_b(b) = 1 and log_b(1) = 0 | log₂(2) = 1, log₂(1) = 0 |
Every one of these is the matching exponent law read backwards. Multiplying numbers adds their exponents, so it adds their logarithms.
A whole-number answer is confirmed by raising the base back to it before it is reported as exact.
A logarithm asks a question about a power: log₂32 is 5 because 2⁵ = 32. Every rule follows from that reading, and a whole-number answer is confirmed by raising the base back to it rather than trusted from a float.
Exact answers verified by raising the base back.
Updated 7 September 2026 · Works in any browser, no installation
log_b(x) asks what power of b gives x. log₂(1024) is 10 because 2¹⁰ = 1024. That reading explains every rule: multiplying numbers adds their exponents, so it adds their logarithms; raising to a power multiplies the exponent, so it multiplies the logarithm. A calculator with only ln and log₁₀ still does every base, because log_b(x) = ln x ÷ ln b.
At a glance
Every logarithm is a question about a power. log₂32 asks: what power of 2 gives 32? The answer is 5, because 2⁵ = 32.
Written that way, most of the confusion disappears. A logarithm is not a new operation so much as the exponent read off the other side of the equation.
This page states the exponential form beside every answer for exactly that reason. If the answer looks wrong, the power is where to check it.
A calculator with only ln and log₁₀ still does every base, because log_b(x) = ln x ÷ ln b.
The base cancels. Both numerator and denominator are measured in the same units, so the ratio does not care which logarithm you used — which is why the natural and common versions above give identical answers.
It also explains why all logarithm graphs have the same shape. They differ only by a constant factor, so log₂ is ln stretched by 1/ln 2 — not a different curve.
A positive base raised to any real power is positive. So nothing you can put in the exponent produces zero or a negative, and no logarithm of them exists.
For zero it is a limit rather than a gap: as the exponent goes more and more negative, the power gets closer to zero without ever arriving. That is why the graph has a vertical asymptote at zero rather than a point.
Complex logarithms of negatives do exist — ln(−1) = iπ — but they are many-valued, since adding 2πi to any answer gives another. That is a genuinely different subject rather than an extension of this one.
ln is base e ≈ 2.71828. It is natural because it is the one whose derivative is 1/x with no constant attached; every other base picks up a factor of ln b, which is the change-of-base division showing up in calculus.
log usually means base 10, and it counts digits: the common log of any whole number is one less than its digit count. That is why it is behind the decibel, the pH scale and the Richter magnitude.
log₂ counts doublings, so it appears wherever something halves or doubles — binary search depth, tree height, bits needed to store a number, entropy.
The notation is not universal. In computer science log often means base 2; in some textbooks it means base e. When it matters, write the base.
log(ab) = log a + log b. Multiplying numbers adds their exponents, and the logarithm IS the exponent. That is the whole derivation.
log(a/b) = log a − log b for the same reason, with division subtracting exponents.
log(aⁿ) = n log a, because raising a power to a power multiplies the exponents.
This is the property that made logarithms indispensable for three centuries: they turn multiplication into addition. Slide rules, log tables and the whole of pre-electronic computation ran on it.
Solving for an exponent. If 2ⁿ = 1000, then n = log₂1000 ≈ 9.97. Any growth question that asks “how long” is a logarithm.
Compressing a range. Sound, earthquakes and acidity all span so many orders of magnitude that a linear scale is useless. Decibels, Richter and pH are all logarithmic for that reason.
Algorithm cost. Halving a problem repeatedly takes log₂n steps, which is why binary search on a million items takes about twenty comparisons rather than a million.
Logarithms are settled; these are the references for the conventions and the named bases.
Method. The value is computed by change of base through the natural logarithm. Before it is reported, the answer is rounded to the nearest integer and the base raised back to that integer: only if the result matches the input is the answer reported as exact. That is why log₂(1024) reads 10 here rather than 9.999999999999998, which is what the division alone produces. That engine is verified on every change against 68 hand-written assertions, including that every whole power of bases 2, 3, 5 and 10 is reported as an exact integer rather than as a decimal that rounds to one. The count and the per-case breakdown are published on the formula verification page.
The inverse operation is the Exponent Calculator, and reading the two side by side is the fastest way to see why the log laws look the way they do. For growth problems specifically, the Doubling Time Calculator answers the commonest one directly.
Educational use disclaimer
This is an educational tool. Exact answers are confirmed by raising the base back to them; everything else is shown to twelve significant figures and is a truncation of an irrational number.
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Published 7 September 2026