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Logarithm Calculator

With number 1000, base 10, logarithm comes to 3.000000 — logarithm. It is reached in 5 steps, the last of which is 6.9077553 / log(10), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Digital Library of Mathematical Functions §4.2, not an approximation fitted to it.

A logarithm in any base, with the natural and base-ten values beside it and the change-of-base step shown.

Formula and sources checked · How we check

Number 1000, Base 10

3.000000

Logarithm for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.

It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.

Logarithm
3.000000
Natural log, ln
log(1000)6.908
Common log, base 10
log10(1000)3
Binary log, base 2
log2(1000)9.966
Log in your base
6.9077553 / log(10)3
Base raised back to the answer
pow(10, 3)1,000

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Worked example

The base-ten log of 1,000 is 3, because 10 raised to the third is 1,000. In base 2 the same number is 9.97, and the change-of-base step is the same division either way: the natural log of 1,000 divided by the natural log of the base.

How to work it out yourself

  1. 1.Enter the number and the base. The number has to be positive — no power of a positive base ever produces zero or a negative.
  2. 2.Read the check line: raising the base to the answer should give back the number you started with, and if it does not, the base was mistyped.
  3. 3.For a base your calculator does not offer, divide two natural logs. That is all change of base is, and it is why log₂ can be computed on a device that only has ln.

The formula

  1. Natural log, lnlog(1000)
  2. Common log, base 10log10(1000)
  3. Binary log, base 2log2(1000)
  4. Log in your base6.9077553 / log(10)
  5. Base raised back to the answerpow(10, 3)

Source: NIST Digital Library of Mathematical Functions §4.2 — powers, roots and logarithms

Questions people actually ask

What is the difference between log and ln?
The base. ln is base e, about 2.71828, and it is the one that appears in growth, decay and compound interest. On most calculators and in most school books "log" means base 10; in mathematical writing and in many programming languages "log" means the natural log. The ambiguity is real enough to have caused published errors, so this page prints both.
What is the change of base formula?
log_b(x) = ln(x) ÷ ln(b), and any base works in place of ln as long as both are the same. It is why a calculator with only two log buttons can produce a logarithm in base 7, and why the answer is a ratio rather than something new.
Why can’t you take the log of a negative number?
Because no real power of a positive base gives a negative result: 10 to anything is positive, however large or small the exponent. The logarithm of a negative number exists in the complex numbers, where it involves π and the imaginary unit, but there is no real answer to return.
What is a logarithm actually for?
Turning multiplication into addition, which is what made slide rules and log tables work. It is also the natural scale for anything that spans orders of magnitude: earthquakes, sound, acidity and stellar brightness are all measured logarithmically because the raw numbers span a range no linear axis can show.

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