How an Antilog Calculator Reads a Log Table Backwards
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
An antilog calculator turns a logarithm back into the number it came from — type 2.4771 and 300 comes back out. Except not quite: you get 299.9853, because 2.4771 is itself log 300 rounded to four decimals. The gap is 0.005% here. Three decimal places later it is 26%, and that asymmetry is the reason this page shows the characteristic, the mantissa and the error alongside the answer instead of one number on its own.
Antilog10(x) is 10x wearing an older name, and the name survives because log tables were built to be read in both directions — forwards to turn a multiplication into an addition, backwards to recover the result. So if it is one exponent, why does anyone get it wrong? Three reasons, and none of them is the arithmetic. People invert the wrong base. They mishandle the integer part of a negative log. And they round the logarithm before raising it, which is where that 26% comes from.
Antilog, Inverse Log and 10ˣ Are the Same Button
Search results treat these as different tools. They are not. Every one of them is “raise the base to this power”, and the only thing that changes between them is which base you meant:
| What people call it | Inverts | Computes | Input 2 gives |
|---|---|---|---|
| Antilog | log₁₀ | 10ˣ | 100 |
| Inverse log | whichever log you used | bˣ | depends on b |
| Inverse ln / exp | ln | eˣ | 7.389 |
| Antilog base 2 | log₂ | 2ˣ | 4 |
Note the gap between rows one and three: 100 against 7.389. Reaching for the wrong one is not a rounding error, it is an answer off by a factor of thirteen. Unqualified, “antilog” means base 10 — that convention comes straight from the printed tables, which were always common logs. If your number came out of a natural logarithm, you want ex instead. And if it came from some other base entirely, the change of base formula converts the exponent before you raise anything.
On a physical calculator there is usually no key with “antilog” printed on it. It is the second function above log, labelled 10x, and the one above ln is ex. Pressing SHIFT then LOG is the antilog; pressing LOG twice is not.
Characteristic Sets the Size, Mantissa Sets the Digits
Here is the split that gives the antilog its shape. Take log₁₀ = 2.4771 and cut it in two: the integer 2 is the characteristic, the fractional 0.4771 is the mantissa. Antilog each piece separately and multiply:
102.4771 = 102 × 100.4771 = 100 × 3.000 = 300.0
That is four-figure working, the way a table would give it. Exactly, 100.4771 is 2.99985 and the answer is 299.9853 — the 3.000 is a rounding, not a fudge, and it is accurate to the four digits the table carries.
The mantissa can only ever produce a number from 1 up to just under 10, because 100 = 1 and 101 = 10. So it supplies the significant digits and nothing else. The characteristic does the rest of the work by sliding the decimal point — which is exactly what scientific notation does, and no coincidence: 3.000 × 10² is the answer in scientific form.
This is why a four-figure log table fits on one page instead of needing a separate volume per order of magnitude. Look at what happens to the mantissa when the number changes size:
| Number | log₁₀ | Characteristic | Mantissa |
|---|---|---|---|
| 3 | 0.4771 | 0 | 0.4771 |
| 300 | 2.4771 | 2 | 0.4771 |
| 30,000 | 4.4771 | 4 | 0.4771 |
| 0.003 | −2.5229 | −3 | 0.4771 |
One mantissa, four numbers. Every value whose digits read 3-0-0 shares it. Reading a table backwards means finding 0.4771 in the body of the table, writing down the digits it points to, and then using the characteristic to decide whether those digits mean 3, 300 or 0.003.
The Bar in a Negative Log Is Not a Minus Sign
Look at that last row again, because it is where most antilog answers go wrong. log₁₀(0.003) is −2.5229. The characteristic is −3 and the mantissa is +0.4771 — different signs, in the same number.
That is not a trick, it is forced. A mantissa has to be positive or it cannot be looked up in a table that only lists 0.0000 to 0.9999. So you take the integer part downwards rather than towards zero: floor(−2.5229) = −3, leaving −3 + 0.4771 = −2.5229. Written in the old notation that is 3.4771, said aloud as “bar three point four seven seven one”.
The overbar covers the 3 and nothing else. It means “this digit alone is negative”. Chop the number towards zero instead and you get a characteristic of −2, which puts the digits 3-0-0 at 0.03 rather than 0.003 — right digits, wrong decade, and the sort of mistake that survives a re-check because the digits look correct.
Chemistry runs into this constantly, because pH = −log₁₀[H⁺] and pKa = −log₁₀Ka, so recovering a concentration or a dissociation constant is always the antilog of a negative number. Take a pKa of 4.7447, near enough acetic acid: Ka = 10−4.7447. Split it into a characteristic of −5 and a mantissa of 0.2553, and since 100.2553 = 1.800 you get Ka = 1.800 × 10−5 — the value acetic acid is normally quoted at. Truncate towards zero instead of flooring and you would report 1.8 × 10−4, an acid ten times stronger than the one in the bottle.
What a Fourth Decimal Is Actually Worth
Logs compress. Antilogs expand — and that includes expanding your error. If your log is off by Δ, the answer is off by a factor of 10Δ, so the relative error is roughly 2.3026 × Δ (that constant is ln 10). Small numbers in, surprisingly large numbers out:
| Log is off by | Answer scales by | Relative error | 300 becomes |
|---|---|---|---|
| ±0.0001 | 1.00023 | ±0.023% | 299.93 – 300.07 |
| ±0.001 | 1.0023 | ±0.23% | 299.31 – 300.69 |
| ±0.01 | 1.0233 | ±2.3% | 293.2 – 307.0 |
| ±0.1 | 1.2589 | ±26% | 238.3 – 377.7 |
Now the four-decimal convention makes sense. Four decimals in the mantissa buys about four significant figures in the answer, which is why every printed table settled on that width and why lab data quotes pH to two decimals and no further — a pH given to 0.01 pins the concentration to about ±2.3%, and the electrode is not more accurate than that anyway.
The practical rule that follows: never round the logarithm before you antilog it. Rounding 2.47714 to 2.48 looks harmless and costs you 0.66% of the answer. Do the whole calculation at full width and round once, at the end.
Four Scales Where Getting the Number Back Means an Antilog
Any measurement quoted on a logarithmic scale needs an antilog to become a physical quantity again. These four cover most of what turns up in coursework, and each one uses a slightly different constant in the exponent:
| Scale | Antilog you need | Worked value |
|---|---|---|
| pH → concentration | [H⁺] = 10^(−pH) | pH 8.1 → 7.94 × 10⁻⁹ mol/L |
| Decibels → power ratio | ratio = 10^(dB/10) | 17 dB → 50.1× |
| Richter → energy | E ≈ 10^(1.5M + 4.8) J | M 6.0 → 6.3 × 10¹³ J |
| Star magnitude → brightness | ratio = 10^(0.4 Δm) | Δm 5 → exactly 100× |
The divisor in the exponent is the part to watch. Decibels divide by 10 for power but by 20 for amplitude, so 17 dB is a power ratio of 50 and a voltage ratio of 7.1. Feed 17 into a plain 10x and you get 1017, which is nonsense of a very confident-looking kind. Set the base to 10 in the calculator above and enter 1.7 rather than 17 — that division is yours to do, not the tool’s.
Where the Digits Are Right and the Answer Still Isn’t
- Splitting a negative log the wrong way. −2.5229 becomes −3 + 0.4771, never −2 + (−0.5229). Both give digits of 3, but the first puts them at 0.003 and the second at 0.03.
- Antilogging an ln with the 10ˣ key. An ln of 3.4012 is 30, not 2,519. The two keys sit next to each other on most scientific calculators and the mistake is nearly silent — the answer looks plausible, it is just wrong by a factor of 84.
- Rounding before the exponent. Covered above, but worth repeating because it is the one error that gets worse the more careful the rest of your work is.
- Forgetting that antilog turns sums into products. antilog(a + b) = antilog(a) × antilog(b). That identity is the whole reason logarithms were invented, and it is the check to run when a multi-step answer looks off: if you added two logs, the antilog must be a product. Working the identity in the other direction is what expanding a logarithm and condensing one are for.
- Assuming the mantissa is the decimal part of the answer. A mantissa of 0.4771 does not mean the answer ends in .4771. It means the digits are 3000 — the mantissa lives in the exponent, not in the result.
What an Antilog Calculator Does When the Answer Will Not Fit
Ask most tools for antilog(500) and they return infinity, because a browser cannot hold a number past about 10308. The characteristic-and-mantissa split walks straight past that limit: the mantissa of 500 is 0.0000, the antilog of that is 1.000, and the characteristic is 500. The answer is 1.000 × 10500, written exactly, with no arithmetic that overflows anything. Try antilog(21.5) and then antilog(500) in the calculator above: the scientific form stays exact through both, while a plain decimal stops being readable at the first and stops existing at the second.
That trick is not a display convenience, it is how logarithms were used for three centuries before electronic calculators: keep the magnitude and the digits in separate columns so neither one can overflow the other. If you want the mechanics of the forward direction, the common logarithm entry sets out the same characteristic and mantissa vocabulary as the tables did. From here, an exponent calculator handles the powers once you have written the answer in scientific form, and a power calculator covers the cases where the exponent is not a logarithm at all.
