Antilog Calculator

Base

Common - base 10

Natural - base e

Binary - base 2

Custom base

Logarithm value (exponent x)

Antilog (result)

Antilog

Characteristic (integer part of x)
Mantissa (fractional part of x)

Split the exponent into its characteristic (2) and mantissa (0). The antilog is the base raised to the characteristic, multiplied by the base raised to the mantissa.

Read the other way, this is a logarithm: the log of 100 for your chosen base is 2.

A positive exponent gives an antilog greater than 1 when the base is above 1.

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Show the growth curve y = b to the x

Plot how the antilog rises as the exponent changes.

Compare the antilog across bases

See the same exponent evaluated in base 10, e, and 2.

The same exponent across common bases
Base	Expression	Antilog of x
Common (10)	10^2	100
Natural (e)	e^2	7.389
Binary (2)	2^2	4

The exponential function reverses the logarithm to give back the original number. If you know that a certain value is the logarithm of x, then the exponential function will raise the base to the power of x and return the original value.

This calculator supports any base. You can select a common base (10), the natural base e or the binary base 2 or enter a custom base. Enter an exponent to get the result and enter a result to find the corresponding exponent.

What is an exponential function?

The exponential function is the inverse of the logarithm. While the logarithm asks "what exponent must the base be raised to in order to get a certain number?", the exponential function starts with the exponent and turns it back into the original base.

So, exponential function of x to base b is equal to b to the power of x.

\operatorname{antilog}_b(x) = b^x

Let's try base 10. The log of 100 to the base 10 is 2, because 10 squared is 100. Conversely, the antilog of 2 is 100.

\log_{10}(100) = 2 \quad\Longleftrightarrow\quad \operatorname{antilog}_{10}(2) = 10^2 = 100

This is how this calculation tool works:

First select the base. The common base (10) is suitable for most everyday calculations, the natural logarithm base (e) corresponds to the ln button on scientific calculators and the binary base (2) is often used in computer science. If you choose a custom base, you can set any positive number as the base but 1 is not allowed.

Next enter the exponent or reciprocal and leave the other field blank. The calculator will calculate the blank field and display the result in that field so both forward and reverse calculations can be done with the same tool.

The formula for the reciprocal:

The entire calculation can be represented by a single formula. The reciprocal is the base raised to the power of and the exponent is the logarithm with base , where the result is the argument of the logarithm.

y = b^x \qquad x = \log_b(y)

The meaning of each symbol is explained below along with examples of pairs.

Symbol	Meaning	Example
b	Base	10
x	Logarithm value (the exponent)	2
y	Antilog (the result)	100

Characteristic and mantissa: The method of logarithmic table

Before the advent of calculators people used log tables to find inverse values. This method can still be used today to explain what the result is made up of. The exponent is divided into an integer part and a fractional part. The integer part is the characteristic, the fractional part is the mantissa.

Since b^x is the product of the power of the characteristic of b and the power of the mantissa of b, the characteristic determines the order of magnitude of the digits while the mantissa determines the leading digit.

b^{x} = b^{\lfloor x \rfloor}\times b^{\,x-\lfloor x \rfloor}

For example, the log of 245 is about 2.389. The characteristic is 2 and the mantissa is .389. So the antilog is 10 squared times 10 to the point 389 which is about 100 times 2.45, back to 245.

The mantissa is always between 0 and 1. This also applies to negative exponents. That's why logarithm tables list not the order of magnitude but only the digits.

Same exponent with different bases.

The inverse function of a number depends entirely on the base. If you substitute exponent 3 into three different common bases, the difference between results will quickly become larger.

Base	Expression	Antilog of 3
Common (10)	10^3	1000
Natural (e)	e^3	20.0855
Binary (2)	2^3	8

Examples of calculations with base 10:

Some examples of the inverse function in base 10 will illustrate this rule. Positive exponents increase the value of a number while negative exponents decrease the value of a number to below 1.

Exponent x	10^x	Antilog
2	10^2	100
0.75	10^0.75	5.6234
-1.5	10^-1.5	0.0316
5.2	10^5.2	158489.32
-3.4	10^-3.4	0.0004

Applications of inverse functions:

When a quantity is expressed logarithmically, one can use the inverse function to determine the actual numerical value.

Chemistry: The pH is the negative logarithm of hydrogen ion concentration, so the concentration is the inverse function of the negative pH. A solution with a pH of 4 has a concentration of 10 to the power of -4 moles per liter.

Acoustics: As sound levels are given in decibels on a logarithmic scale, the inverse function must be calculated to obtain the actual intensity ratio.

Earthquakes: The magnitude on the Richter scale is given in a logarithmic way so that the ratio of energy between two magnitudes is the inverse function of their difference.

Growth: When a compounded growth is linearized by a logarithmic transformation, the actual rate of growth or ending amount can be recovered using the reciprocal.

What does that graph look like?

The reciprocal curve y = b^x has the same shape for any positive value of b that is greater than 1. As the exponent approaches negative infinity, the curve approaches 0 but never touches the axes. As the exponent becomes larger, the curve increases exponentially.

All of these curves go through the same point. When the exponent is 0, the corresponding value is 1. This is because any base raised to the power of 0 is equal to 1. The tool's growth curve plots this point exactly around the entered exponent.

Hints and side conditions:

The value for the base must be a positive number and cannot be 1 because any power of 1 is always 1.

For natural logarithms, use the predefined value for e and do not enter a value of 2.718. This will allow you to take full advantage of the precision of e.

Large exponents result in rapid growth. The equivalent of 20 is 10 to the power of 20 when the base is 10. So it can happen that the result will be shown in scientific notation.

As long as the base is a positive number, negative exponents will always result in a positive number between 0 and 1. It can never be less than 0.

Frequently asked questions

What is the difference between an antipode and an exponent?

These are different explanations of the same calculation, just from different perspectives. The antilog of x in base b is b raised to x, exactly like an exponent. The term antilogarithm indicates that a logarithm is reversed and exponent is the general term for raising a base to a certain power.

Is Ln an antilog?

The function ln is the natural logarithm, i.e. a logarithm with base e, where e is approximately 2.71828. The antilogarithm is the inverse operation of raising a base to an exponent. The antilogarithm that corresponds to ln is e raised to x. That's why the keypad for "e to the x" on a calculator calculates the natural antilog.

Can an antilog be negative?

With positive bases and real exponents, the result cannot be negative. Any power of a positive base remains positive. The exponent entered can also be negative. In this case, the antilogarithm is between 0 and 1 and never less than 0.

How do you calculate antilog without a calculator?

For simple numbers, one can calculate mentally. The reciprocal of 3 is 1000 if the base is 10 and 8 if the base is 2. For more complex powers, logarithm tables were used in the past. In this case, the exponent is divided into a characteristic number and a mantissa number and then the numbers are read from the table.

What is the reciprocal of 3?

It depends on the base. If the base is 10, then the reciprocal of 3 is 1000, if the base is 2 it's 8, and if the base is e it's about 20.09. A calculator shows these three results simultaneously in a table comparing bases.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

GraphPad: Review of Logarithms and Antilogarithms
How logs and antilogs invert each other across bases.
NIST Special Publication 811: Guide for the Use of the International System of Units
The US national standard for SI units, unit names and symbols, and conversion factors.