Antilog Calculator - Inverse Logarithm Any Base
Calculate the antilogarithm (inverse logarithm) for any base: base 10, natural log (e), base 2, or any custom base. Instant results with formula.
Enter the base and logarithm value, then click Calculate to get the antilogarithm result. The formula is: antilog_b(y) = b^y.
Antilog Calculator - Inverse Logarithm Any Base
Calculate the antilogarithm (inverse logarithm) for any base: base 10, natural log (e), base 2, or any custom base. Instant results with formula.
About the antilog calculator
Antilogarithm, commonly abbreviated as antilog, is the inverse operation of the logarithm. Just as the square root undoes squaring and division undoes multiplication, the antilogarithm undoes the logarithm. If log_b(x) = y, then x = antilog_b(y) = b^y, where b is the base, y is the logarithm value (also called the mantissa plus characteristic in older conventions), and x is the original number being recovered.
The relationship is elegantly simple: taking the antilog of a value y with base b means raising b to the power y. For common logarithms (base 10), antilog₁₀(y) = 10^y. For natural logarithms (base e ≈ 2.71828), the antilog is the exponential function e^y. For binary logarithms (base 2), antilog₂(y) = 2^y. This generalises to any valid base: a base must be a positive real number different from 1, ensuring that the logarithm function it inverts is well-defined and monotone.
Antilogarithms are essential in any field where logarithmic scales are used to compress data spanning many orders of magnitude. In chemistry, pH is defined as −log₁₀([H⁺]), where [H⁺] is the molar concentration of hydrogen ions. To recover the concentration from a pH reading, you apply the antilog: [H⁺] = 10^(−pH). A pH of 3 corresponds to [H⁺] = 10^(−3) = 0.001 mol/L, and a pH of 7 corresponds to [H⁺] = 10^(−7) = 0.0000001 mol/L. The antilog makes the multiplicative difference between these concentrations immediately apparent.
In seismology, the Richter scale is logarithmic: a magnitude-7 earthquake releases about 31.6 times more energy than a magnitude-6 earthquake because 10^(7/1.5) / 10^(6/1.5) ≈ 31.6. Converting from magnitude to energy release requires antilogarithms. Similarly, the decibel scale for sound intensity defines L = 10 × log₁₀(I/I₀), so recovering intensity from decibels requires I = I₀ × 10^(L/10). Financial mathematics uses the natural antilog e^(rt) to compute continuously compounded interest, where r is the annual rate and t is the time in years.
A frequent misconception is that antilog is the reciprocal of log: antilog_b(y) ≠ 1/log_b(y). The antilog is the inverse function (exponentiation), not the multiplicative inverse. For example, log₁₀(100) = 2, so antilog₁₀(2) = 100, not 0.5. Another important point is that the base matters: antilog₁₀(3) = 1000 but antilog₂(3) = 8. Always match the base of your antilog to the base that was used in the original logarithm.
This calculator also supports the antilog properties: the addition rule antilog_b(y₁ + y₂) = antilog_b(y₁) × antilog_b(y₂) is equivalent to b^(y₁+y₂) = b^y₁ × b^y₂, which is just the exponent product rule. These properties allow you to multiply large numbers by adding their logarithms and then taking the antilog of the sum, which is the principle behind the historical use of logarithm tables before electronic calculators were available.
Antilog examples
Common antilogarithm calculations across different bases and applications.
| Base / Log Value | Antilog Result | Application |
|---|---|---|
| b=10, y=2 | 100 | 10² = 100. Used in pH calculation: a pH of 2 means [H⁺] = 10⁻² = 0.01 mol/L. |
| b=e, y=1 | ≈2.71828 | e¹ = e ≈ 2.71828. The natural antilog of 1 is Euler's number itself. |
| b=2, y=8 | 256 | 2⁸ = 256. Binary antilog: 8 bits can represent 256 distinct values. |
| b=10, y=−3 | 0.001 | 10⁻³ = 0.001. pH of 3 gives hydrogen ion concentration of 0.001 mol/L. |
How to use the antilog calculator
- Select a common base using the quick-select buttons (Base 10, Base e, Base 2), or choose 'Custom Base' and type any valid base value into the Base field.
- Enter the logarithm value (y) in the Logarithm Value field. This can be any real number, including decimals and negative values.
- Click Calculate. The antilog result (b^y) appears instantly along with the formula and a verification that log_b(result) ≈ y.
- For pH calculations use Base 10 and enter the negative pH as the log value: pH 4 → y = −4 → result = 0.0001 mol/L.
- Click Reset to clear all fields and start a new calculation.
Antilog calculator FAQ
What is the antilog of a number?
The antilogarithm of a number y with base b is b raised to the power y: antilog_b(y) = b^y. It is the inverse operation of the logarithm. If log_b(x) = y, then antilog_b(y) = x, recovering the original value x from its logarithm y.
Is antilog the same as 10^x?
Antilog₁₀(x) = 10^x, so for base-10 logarithms the two expressions are identical. However, antilog is a general concept that works with any base: antilog_e(x) = e^x, antilog₂(x) = 2^x, and antilog_b(x) = b^x. When people say 'antilog' without specifying a base, they usually mean base 10.
How do I use antilog for pH calculations?
pH is defined as −log₁₀([H⁺]), where [H⁺] is the hydrogen ion concentration in moles per litre. To find the concentration from the pH, rearrange to get [H⁺] = 10^(−pH). For example, pH 4 gives [H⁺] = 10^(−4) = 0.0001 mol/L. In this calculator, set Base = 10 and Logarithm Value = −pH.
What is the antilog of a negative number?
The antilog of a negative number is a positive number less than 1 (for bases greater than 1). For example, antilog₁₀(−2) = 10^(−2) = 0.01. Negative log values simply mean the original number was a small fraction. All antilog results are positive because any positive base raised to any real power is always positive.
What base should I use for natural antilog?
For the natural antilog, use base e ≈ 2.71828. This corresponds to the exponential function exp(y) = e^y. Natural antilogs appear in continuous compound interest (A = Pe^(rt)), radioactive decay (N = N₀e^(−λt)), population growth, and many areas of physics. In this calculator, click the 'Base e' button and it will set the base automatically.
What are the domain restrictions for antilog?
The base b must be a positive real number not equal to 1. There is no restriction on the logarithm value y — it can be any real number, positive, negative, or zero. The result antilog_b(y) = b^y is always a positive real number for valid inputs. Antilog₁₀(0) = 10⁰ = 1, which is the neutral value for the common logarithm scale.