Antilog Calculator

What is an antilogarithm?

An antilogarithm, or antilog, is the number you get back after undoing a logarithm. A logarithm answers the question "which exponent turns this base into that number?", and the antilog runs the question in reverse: give it the exponent and it returns the number. Saying that the logarithm of x to base b equals y is the same as saying that b raised to the power y equals x, so the antilog of y is just b to the power y.

antilog_b(y) = b^yIf log_b(x) = y, then antilog_b(y) = x.

Two bases account for most uses. With base 10 (the common logarithm) the antilog of y is 10^y. With base e ≈ 2.71828 (the natural logarithm) it is e^y, better known as the exponential function. Base 2 appears in computing and information theory. On a handheld calculator the antilog usually sits above the log or ln key and is reached with a shift or inverse key, labelled 10^x or e^x.

How to use the antilog calculator

1. Type the logarithm value into Number. It may be positive, negative or a decimal.
2. Type the Logarithm base. Leave it empty to use the natural logarithm, base e. A base of zero or below is treated the same way, because logarithms do not exist for those bases.
3. Press Calculate. The result is the base raised to your number.

Results are rounded to six decimal places. Whole-number answers appear without decimals (base 10 and number 2 gives 100), thousands are separated with commas, and other answers are padded to two, four or six decimals depending on how many digits they need: 0.5 appears as 0.50, 0.125 as 0.1250 and the square root of ten as 3.162278. An exponent big enough to overflow is reported as inf.

Worked examples

Base 10. The common antilog of 2.5 is 10^2.5 = 10² × 10^0.5 = 100 × 3.162278 = 316.227766. Enter base 10 and number 2.5 to confirm it.

Base 2. The antilog to base 2 of 5 is 2 × 2 × 2 × 2 × 2 = 32, the number of different values that five binary digits can hold.

Natural antilog. With the base left empty, the antilog of 1.2 is e^1.2 = 3.320117.

Negative values. A negative logarithm gives an answer between 0 and 1. The antilog to base 10 of −3 is 1 ÷ 10³ = 0.001, and the antilog to base 2 of −3 is 1 ÷ 8 = 0.125.

Antilogs with log tables: characteristic and mantissa

Before calculators, antilogs were read from printed tables. A base-10 logarithm is split into a characteristic, its whole-number part, and a mantissa, its decimal part. The mantissa fixes the digits of the answer and the characteristic fixes where the decimal point goes.

Example. To find the antilog of 2.6021, look up the mantissa 0.6021. It corresponds to the digits 4.000 (10^0.6021 = 4.0004). The characteristic 2 moves the decimal point two places to the right, so the answer is about 400. Keep the same mantissa with a characteristic of −1 and you get about 0.4.

Mind the rounding. Because the exponent sits in a power, a small error in y is magnified in the answer: log 3.45 is 0.537819..., and rounding it to 0.5378 returns 3.4498 instead of 3.45.

Where antilogs are used

• Chemistry (pH): hydrogen-ion concentration in mol/L is 10^−pH. A solution with pH 3.5 has 10^−3.5 ≈ 0.000316 mol/L.
• Sound (decibels): a power ratio equals 10^{dB ÷ 10}, so a rise of 30 dB means 10³ = 1,000 times the power.
• Earthquakes: magnitude is logarithmic, and each whole step corresponds to ten times the ground motion. A magnitude 6 quake moves the ground 10² = 100 times as much as a magnitude 4.
• Continuous growth: a balance growing at 5% a year, compounded continuously, multiplies by e^{0.05 × 10} = e^0.5 ≈ 1.648721 over ten years, a gain of about 64.9%.
• Computing: n bits can represent 2ⁿ different values; the antilog to base 2 of 10 is 1,024.

To go in the opposite direction and find the exponent, use the log calculator. For ordinary powers with a whole-number exponent the exponent calculator is a quicker route.

Common mistakes

• Using the wrong base. A logarithm taken to base 10 must be undone with base 10. Undoing it with base e gives a different number: e² ≈ 7.389056, not 100.
• Mixing up log and ln. On most calculators "log" means base 10 and "ln" means base e, although some textbooks use "log" for base e.
• Expecting a negative answer. A positive base raised to any real power is positive, so an antilog is never negative.
• Rounding the exponent too early. Keep as many digits of the logarithm as you can until the final step.

A base of 1 always returns 1, because 1 raised to any power is 1. Very large exponents overflow: 10¹⁰⁰⁰ is beyond what a standard double-precision number can hold, so the calculator shows inf.