Antilog Calculator
Enter a logarithm value and choose a base to calculate its antilogarithm. The antilog of y in base b is simply b raised to the power y. This calculator supports base 10 (common log), base e (natural log), base 2 (binary log), and any custom base. Switch to reverse mode to find the log value from a known result.
What is an antilogarithm?
An antilogarithm is the inverse of a logarithm. If log_b(x) = y means "b raised to y gives x," then the antilog of y in base b is simply b^y, which gives back x. For example, log_10(100) = 2, so antilog_10(2) = 10^2 = 100. Every logarithm has a corresponding antilog: antilog reverses the compression that logarithms apply to large or small numbers. Before electronic calculators, scientists used printed log tables to multiply large numbers by adding their logs and then looking up the antilog of the sum.
Antilog formula and how to calculate it
The formula is: antilog_b(y) = b^y. For base 10, use 10^y. For the natural antilog (base e), use e^y, which is also written exp(y). For base 2, use 2^y. To find the antilog of a value from a logarithm table, you identify the characteristic (integer part of the log) and the mantissa (decimal part), look up the mantissa in the antilog table, and then apply the characteristic as a power of 10 to position the decimal point. Modern calculators skip the table entirely: just raise the base to the given exponent.
Base 10, natural log, and base 2 antilogs
Base 10 (common log) antilogs appear in chemistry (pH = -log[H+], so [H+] = 10^-pH), acoustics (decibels use a base-10 scale), and seismology (the Richter scale). Natural antilogs (base e = 2.71828) are central to continuous growth and decay: compound interest, radioactive half-life, and population models all use e^y. Base 2 antilogs are the backbone of computer science: 2^n bytes can store 2^n distinct values, and binary search trees split data by powers of 2. The change-of-base formula relates all bases: log_b(x) = ln(x) / ln(b), so antilog_b(y) = e^(y * ln(b)).
Real-world applications of antilogs
Antilogs bridge the logarithmic world back to the linear world. In chemistry, the hydrogen ion concentration of a solution is 10^-pH: a pH of 3 means [H+] = 10^-3 = 0.001 mol/L. In sound engineering, a sound level of 80 dB corresponds to an intensity of 10^8 times the reference intensity. In finance, a continuously compounded return r over time t gives a growth factor of e^(r*t). In information theory, the number of distinct messages encodable with n bits is 2^n. In astronomy, the brightness ratio between two stars differing by m magnitudes is 10^(m/2.5).
Common antilog values (base 10)
| y (log value) | Antilog base 10 (10^y) | Antilog base e (e^y) | Antilog base 2 (2^y) |
|---|---|---|---|
| -3 | 0.001 | 0.04979 | 0.125 |
| -2 | 0.01 | 0.13534 | 0.25 |
| -1 | 0.1 | 0.36788 | 0.5 |
| 0 | 1 | 1 | 1 |
| 0.5 | 3.16228 | 1.64872 | 1.41421 |
| 1 | 10 | 2.71828 | 2 |
| 1.5 | 31.6228 | 4.48169 | 2.82843 |
| 2 | 100 | 7.38906 | 4 |
| 3 | 1000 | 20.0855 | 8 |
| 4 | 10,000 | 54.5982 | 16 |
| 5 | 100,000 | 148.413 | 32 |
| 10 | 10,000,000,000 | 22026.5 | 1024 |
These are the standard values used in logarithm tables and science calculations.
Frequently asked questions
What is the antilog of 0 for any base?
Any number raised to the power 0 equals 1, so the antilog of 0 is always 1, regardless of the base. For example, 10^0 = 1, e^0 = 1, and 2^0 = 1. This is a fundamental property of exponentiation.
What is the antilog of a negative number?
For any base greater than 1, the antilog of a negative number is a positive fraction between 0 and 1. For example, antilog_10(-2) = 10^-2 = 0.01, and antilog_e(-1) = e^-1 = 0.3679. Antilogs are never negative or zero, because a positive base raised to any real power is always positive.
How do I calculate the natural antilog (base e)?
The natural antilog is e^y, also written exp(y). On a scientific calculator, press the e^x key (or "EXP" on some models). For example, the natural antilog of 2 is e^2 = 7.38906. In Python, use math.exp(2); in Excel or Google Sheets, use =EXP(2).
What is the difference between antilog and inverse log?
They are the same thing. "Antilog" and "inverse logarithm" both refer to the operation that undoes a logarithm, which is raising the base to the given power. The notation log^-1(y) or antilog(y) both mean b^y for the relevant base b.
How do I find the antilog from a log table?
Separate your log value into its characteristic (the integer part) and mantissa (the decimal part). Look up the mantissa row and column in the antilog table to get a four-digit number. Then multiply that number by 10 raised to the characteristic. For example, log = 2.3010: mantissa 0.3010 gives 2.000 from the table, characteristic 2 means multiply by 10^2 = 100, giving 200. A calculator is faster and more accurate, but understanding the table method helps you see what antilogs really do.
Can the base of an antilog be negative or a fraction?
The base must be a positive number other than 1 for standard real-number logarithms. Negative bases and base 1 make the logarithm undefined or trivial. Fractional bases (like 0.5) are valid: antilog_0.5(3) = 0.5^3 = 0.125. For bases between 0 and 1, the antilog decreases as y increases instead of growing.