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Free Online Logarithm Calculator, Natural Log, Log Base 10 and Any Base

Work out a logarithm in any base, including the natural log and log base ten, or go the other way from a logarithm back to the number. The answer comes with the power it represents written out, so you can check it, and the awkward cases of zero, one and negative numbers are explained rather than refused.

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Free Online Logarithm Calculator, Natural Log, Log Base 10 and Any Base

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📉 Logarithm Calculator Any base · natural log · and the way back

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📉 The logarithm

A logarithm answers one question: what power of the base gives this number?

Base

Or type a base Write e for the natural log. Any base above zero works, apart from one.

The number Has to be above zero. Below one gives a negative logarithm.

Try one of these

Logarithm

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Written as a power—

Natural log, ln—

Base 10, log—

Base 2, log2—

Recent logarithms, kept in this browser only.

Nothing here yet Press Log to history on the first tab to keep a calculation here.

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How to Work Out a Logarithm

Four steps, and the third is the one that makes it stick.

Choose the base, and be explicit about it Ten, two and e are one tap away. The word log on its own means different bases in different fields, which is the single biggest cause of confusion, so the base is always written out here.

Type the number It has to be above zero. A number between zero and one gives a negative logarithm, which is correct rather than an error.

Read the answer as a power Underneath the answer the same fact is written as a power, so log base ten of 1000 being 3 appears as ten cubed equals 1000. That is the definition, and seeing it is better than remembering it.

Or work backwards The switch at the top turns the question round: give it a logarithm and it gives you the number. That is the antilogarithm, and it is just the base raised to that power.

What to Know About Logarithms

What log means without a base, and why some inputs have no answer.

The word log on its own means three different things. In school mathematics it means base ten. In computer science it usually means base two. In most programming languages and in higher mathematics it means the natural log, base e. These are not small differences: log of 1000 is 3, about 9.97, or about 6.91 depending on which one you meant. The base is written out on every line here for that reason.

A negative number has no logarithm among real numbers. A positive base multiplied by itself any number of times, including a fractional number of times, is always positive. So no power of ten gives minus five, and the logarithm of minus five does not exist on the number line. It does exist among complex numbers. This is a fact about numbers rather than a limitation of the page.

The logarithm of zero is not minus infinity, it is undefined. As the number gets closer to zero the logarithm falls without limit, so people write minus infinity as a shorthand for that behaviour. But there is no power of ten that equals zero, so there is no answer, and saying minus infinity invites you to treat it as a number in a later step, which goes wrong.

Base one never has an answer, and base zero is worse. One multiplied by itself is always one, so there is no power of one that gives five, and every power of one gives one, so even the logarithm of one in base one is not a single answer. A calculator that returns infinity here is giving you a number where there is none.

A whole number answer is recognised and checked. Log base two of 1024 is exactly ten, and the page says so rather than showing 9.999999999999998. The recognition is not a guess: the base is raised back to the rounded answer and only kept if the original number comes back out.

Any base is worked out from the natural log, and that is fine. The log of x in base b is ln x divided by ln b. Browsers have accurate natural, base ten and base two logarithms built in, so those three are used directly, and anything else goes through the division. The difference is in the last digit or two, never in anything you would notice.

Key Features & Capabilities

What it does, and what it deliberately does not.

Any base, stated plainly Ten, two and e one tap away, anything else typed in, and the base written into every line of the answer.

Both directions Number to logarithm, or logarithm back to number, which is the antilogarithm, on the same page.

The answer as a power The same fact written the other way round, which is the definition of a logarithm and makes it stick.

Three bases at once Natural, base ten and base two alongside your own base, because people often need to quote more than one.

Honest about what has no answer Zero, negatives and base one are explained rather than given a plausible looking number.

Nothing is uploaded It all runs in this tab and keeps working with the network off.

About Logarithms

A logarithm answers one question, and it is worth having the question in front of you: what power of the base gives this number? Log base ten of a thousand is three because ten cubed is a thousand. That is the whole idea, and every rule about logarithms follows from it. So the answer here always comes with the power written out, because the definition read back is more useful than the definition recalled.

The practical difficulty with logarithms is not the arithmetic, it is the notation. Write log with no base and three different fields will read it three different ways: base ten in school mathematics, base two in computing, and base e in most programming languages and in anything beyond school level. The answers are not close to each other. The logarithm of a thousand is three, or about nine point nine seven, or about six point nine one. A tool that prints log without saying which is inviting the mistake, so the base appears on every line here.

The second difficulty is the inputs that have no answer, and there are three of them. A negative number has no real logarithm, because no positive base raised to any real power comes out negative. Zero has none either, although the logarithm falls without limit as you approach zero, which is why people write minus infinity and why writing it is unhelpful: it looks like a number and will be treated like one in the next step. And base one has no logarithms at all, because one to any power is one. Returning infinity for these, which several calculators do, hands you something that will quietly poison whatever you do next.

The rest is making the result usable. A whole number answer is shown as a whole number, verified by raising the base back to it rather than by rounding and hoping. The natural, base ten and base two logarithms are given alongside your own base, since quoting more than one is common. And going backwards, from a logarithm to the number it came from, is the same page rather than a different tool, because it is the same relationship read in the other direction.

Frequently Asked Questions

Bases, natural logs, and the cases with no answer.

It depends entirely on who wrote it. School mathematics means base ten, computer science usually means base two, and most programming languages and higher mathematics mean base e. That is why the base is written out on every line here instead of being left to convention.

The logarithm in base e, where e is about 2.71828. It turns up wherever something grows or decays continuously, which is why it is called natural, and it is written ln. Type e as the base and you get it.

Because no positive base raised to any real power gives a negative result. The answer exists among complex numbers but not on the number line, so there is nothing real to report. This is a property of numbers, not a shortcoming of the calculator.

There is not one. The logarithm falls without limit as the number approaches zero, which people write as minus infinity, but no power of the base actually equals zero. Treating minus infinity as a number in a later calculation is where that shorthand causes trouble.

Because one multiplied by itself is always one, so no power of one gives any other number, and every power of one gives one. There is no answer to return, and returning infinity would be inventing one.

Use the switch at the top of the panel. That is the antilogarithm, and it is simply the base raised to the power of the logarithm. If the logarithm was 3 in base ten, the number is a thousand.

No. The whole calculation happens in your browser, so nothing leaves the page and it keeps working offline once loaded.