Antilog Calculator
The antilog of 3 in base 10 is 1,000, because 10 raised to the power 3 is 1,000. This antilog calculator reverses a logarithm, raising the base to your number instead of finding an exponent. It handles the common base 10, the natural base e, base 2, or any custom positive base you type in, and it accepts negative, decimal, and zero inputs. Antilogs come up whenever a log transform needs to be undone, such as converting a pH value back to hydrogen ion concentration, turning a decibel or Richter figure back into a raw quantity, reversing a logarithmic regression result, or simply checking log table work by hand. Enter the value, choose the base that matches the logarithm you are reversing, and the calculator returns the antilog instantly, switching to scientific notation automatically when the result is very large or very small.
Quick answer
A logarithm asks what exponent produces a number. The antilog runs that in reverse, applying the exponent to the base.
Antilog
1,000
What this tells you
- •A logarithm asks what exponent produces a number. The antilog runs that in reverse, applying the exponent to the base.
- •Antilog of x in base b is simply b to the power x, written as b^x.
- •Antilog base 10 of 2 is 100, of 3 is 1000, and of -2 is 0.01.
- •Taking the antilog of a log returns the original number, which is how you undo a log transform.
- •The base must be a positive number other than 1, since base 1 has no working logarithm or antilog.
How to Use
- 1Enter the number whose antilog you want. This is usually a log value you already calculated or measured and now need to reverse.
- 2Enter the base. Use 10 for common logs, 2.71828 for natural logs, or 2 for binary logs used in computing.
- 3Click Calculate to raise the base to the power of your entered value.
- 4Read the antilog as the main result. It is always positive, since a real base raised to any real power cannot produce a negative or zero result.
- 5Very large or very small results switch to scientific notation automatically so long strings of digits stay readable.
How It Works
Formula
antilog_b(x) = b^xRaise the base to the power of your value. The antilog base 10 of 2.5 is 10^2.5, about 316.23, which sits between 10^2 = 100 and 10^3 = 1000 because 2.5 sits between 2 and 3. Negative inputs give values below 1, since 10^-2 equals 1 divided by 10^2, or 0.01. A value of exactly 0 always returns 1, because any positive base raised to the power of 0 equals 1. This is exactly how log tables were used in reverse before calculators: find the log of two numbers, add them together, then take the antilog of the sum to multiply the original numbers.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Antilog of 3, base 10
10 x 10 x 10 = 1,000. This is the most common antilog check, since base 10 logs appear throughout science and engineering.
Antilog of 2.5, base 10
10^2.5 sits between 10^2 = 100 and 10^3 = 1000. The exact value is 316.227766, which rounds to about 316.23 for everyday use.
Antilog of 1, base e
e^1 is e itself, so the natural antilog of 1 returns the base value used for the calculation.
Antilog of -2, base 10
A negative exponent produces a reciprocal, so 10^-2 equals 1 divided by 100, which is 0.01. The result stays positive even though the input is negative.
Antilog of 3, base 2
2^3 equals 8. Using the same input value with a different base produces a completely different result, which is why matching the base to the original log is essential.
Base 10 Antilog Table
Common log values and their base 10 antilogs.
| x | antilog(x) |
|---|---|
| -2 | 0.01 |
| -1 | 0.1 |
| 0 | 1 |
| 0.5 | 3.1623 |
| 1 | 10 |
| 2 | 100 |
| 3 | 1000 |
| 6 | 1000000 |
How antilogs and logs work as a pair
A logarithm and an antilog are inverse operations built around the same base. If log base 10 of 1000 is 3, then the antilog base 10 of 3 is 1000. Running one operation and then the other, in either order, returns the number you started with, which makes antilogs useful for checking log calculations by hand.
Before calculators had built-in log and power functions, this relationship was the basis of log tables: converting multiplication into addition by taking logs, adding the results, then reading the antilog of the sum to get the final product. The same base-matching rule still applies today. A base 10 antilog will not reverse a natural log, and a natural antilog will not reverse a base 10 log.
Common mistakes
- Using the wrong base. The antilog of 3 is 1000 in base 10 but only 8 in base 2. The base must match the log you are reversing.
- Confusing antilog with the reciprocal of a log. Antilog(x) is b^x, not 1/log(x), and the two are unrelated operations.
- Dropping the sign. Antilog of -2 in base 10 is 0.01, not -100. Negative exponents shrink the result, they do not negate it.
- Entering the natural base as a rounded 2.7 instead of a more precise value like 2.71828, which introduces small errors for larger exponents.
- Expecting a negative or zero antilog result. Since a positive base raised to any real power stays positive, the antilog is always greater than zero.
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