Antilog Calculator

What this calculator does

So, you have a logarithm value and you want the number it came from. This tool finds the antilogarithm: give it a logarithm and a base, and it returns the original number, to two decimal places.

There are two inputs, the logarithm value and the base, and the base starts at 10 unless you change it. It is the tool that runs a logarithm backwards.

How to use it

1. Enter the logarithm value (y).
2. Enter the base (b), or leave it at 10.
3. Press Calculate.

The base must be positive. The logarithm value itself can be any number, positive, negative, or zero.

What an antilogarithm is

An antilogarithm is simply the base raised to a power. The antilog of a value y in base b is b raised to the power y. So the antilog of 3 in base 10 is 10 to the power 3, which is 1000. In other words, taking an antilog is the ordinary act of raising the base to a power; the name just signals that you are doing it to undo a logarithm rather than for its own sake. That is exactly what the tool computes: the base to the power of the value you entered.

The exact reverse of a logarithm

Logarithm and antilogarithm are a matched pair, each undoing the other. A logarithm takes a number and asks what power the base was raised to; an antilogarithm takes that power and gives back the number. So if the log base 10 of 1000 is 3, then the antilog base 10 of 3 is 1000, right back where you started. Feed a number through a log and then through an antilog in the same base, and you return to the original. This is the same relationship that raising to a power has with taking a logarithm, seen from the other side.

The base, and the default of 10

The base you choose must match the base of the logarithm you are reversing. If your value came from a common logarithm, use base 10; from a natural logarithm, use base e; from a binary logarithm, use base 2. The tool defaults to base 10, since common logarithms are the most frequent, but you can set any positive base. Getting the base right matters, because the same logarithm value gives very different numbers in different bases: an antilog of 3 is 1000 in base 10, but only 8 in base 2.

A worked example

Enter a value of 3 with the base left at 10. The tool returns 1000, since 10 to the power 3 is 1000. Change the base to 2 and it returns 8, because 2 cubed is 8. Enter a value of 2 in base 10 and it returns 100. A negative value works too: an antilog of minus 1 in base 10 is 0.1, since 10 to the power minus 1 is one tenth.

Why antilogs existed

The antilogarithm earned its own name in the age of log tables. To multiply two awkward numbers, people would look up the logarithm of each, add the logs together, and then take the antilog of the total to get the answer, converting a hard multiplication into an easy addition. Antilog tables, which ran that final step, sat alongside log tables in every set of mathematical tables. The calculators changed the tools, but the operation is unchanged: an antilog is still a base raised to a power.

Questions people ask

What is an antilogarithm?

The base raised to a power. The antilog of y in base b is b to the power y, which is the number whose logarithm is y.

How does it relate to a logarithm?

It is the exact reverse. A log turns a number into a power; an antilog turns that power back into the number.

Which base should I use?

The same base as the logarithm you are undoing: 10 for common logs, e for natural logs, 2 for binary logs. The tool defaults to 10.

Can the logarithm value be negative?

Yes. A negative value gives a result between 0 and 1. The antilog of minus 1 in base 10 is 0.1.

How do I go the other way?

Take a logarithm. The general logarithm calculator, and the base 2, base 10, and natural log tools, all go from a number to its logarithm.

References

On the antilogarithm. The antilog is the inverse of the logarithm, found by raising the base to the given power, so that log and antilog in the same base undo each other.

1. Eric W. Weisstein, "Antilogarithm," from MathWorld, a Wolfram resource, on the antilogarithm as the inverse of the logarithm, equal to the base raised to the power.
2. Eric W. Weisstein, "Logarithm," from MathWorld, a Wolfram resource, on the logarithm, the forward function the antilogarithm reverses.