What does antilog actually mean?

It means raising the base to that power. antilog₍b₎(y) = bʸ, so the antilog of 3 in base 10 is 1000, and that is the entire definition. It is the inverse of taking a logarithm, which is why the two undo each other: the logarithm of the antilog of y is y again. There is no separate operation to learn and no special key required — anything that can raise a number to a power can take an antilog.

If it is just exponentiation, why does the word survive?

Because of how tables were used, and because several fields still quote logarithms as their everyday units. A book of logarithms printed only the fractional part of each value, since the whole part merely placed the decimal point. You looked up two logarithms, added them to multiply, then went back through a separate table of antilogarithms to recover the digits. Slide rules did the same thing mechanically. The tables are gone, but pH, decibels and earthquake magnitudes are all logarithms by definition, so "take the antilog" remains something people say and mean.

How do I take an antilog on a calculator or in code?

On a calculator it is usually the second function above the log key, printed as 10ˣ, and above ln you will find eˣ. In code there is no antilog function at all: use pow(base, y), or exp(y) when the base is e. Watch the notation trap — in most programming languages log means the natural logarithm, so its inverse is exp rather than the 10ˣ your calculator gives you. If you are working from a value labelled just "log", check which base it came from before undoing it.

What are the characteristic and the mantissa?

The characteristic is the whole part of a logarithm and the mantissa is the fraction. The characteristic tells you only the size of the answer — it becomes a clean power of the base — while the mantissa supplies every significant digit. That is why a table of logarithms could print the fraction alone and let you supply the rest, and it is why log 3, log 30 and log 3000 share the same digits after the decimal point. Note the convention for negatives: the characteristic is the floor, so a logarithm of −2.5 splits into −3 and 0.5, not −2 and −0.5.

Why does rounding a logarithm lose so much precision?

Because the antilog turns an absolute error into a relative one. If your exponent is wrong by d, the answer is wrong by a factor of roughly ln(base) × d — so in base 10, an error of 0.00005 in the exponent becomes 0.0115 per cent of the answer, whatever the answer happens to be. That is the same fractional error whether the result is 3 or 3 billion, which is what makes it easy to underestimate. A logarithm quoted to four decimals simply does not contain more than about four significant figures, however many the display shows afterwards.

How many decimal places should I keep in a logarithm?

One more than the number of significant figures you need in the answer, near enough. The relationship is exact and simple: every extra decimal place in the exponent buys precisely one more digit in the result, because a factor of ten in the error becomes a factor of ten in the relative error. Four decimals gives just under four significant figures in base 10, five gives just under five, and so on. The practical rule that follows is to round only at the very end — carry the logarithm at full precision through any intermediate arithmetic and antilog once.

Why does the antilog overflow so easily?

Because logarithms compress an enormous range into a small one, and undoing that runs out of room fast. Every number a double can hold has a base-10 logarithm between about −324 and 309, so any logarithm you meet fits easily. Go the other way and the window is only about 632 wide: above y = 308.2547 the answer is infinity and below y = −323.6072 it is zero. In base e the limits are 709.7827 and −745.1332. This one-sided failure is why probabilities and likelihoods are added as logs and only converted back at the end, if at all.

How do I convert pH back to concentration, or decibels to a ratio?

A pH is a negative base-10 logarithm, so the concentration of hydrogen ions is 10^(−pH): a pH of 3 means 10⁻³, or 0.001 moles per litre. Decibels need care about which quantity you have, since a power ratio is 10^(dB/10) while an amplitude or voltage ratio is 10^(dB/20) — the factor of two is where most mistakes happen. Earthquake magnitudes are messier still: each whole step is ten times the ground motion but about 31.6 times the energy, since energy goes as 10^(1.5M).

What is the antilog of a negative number?

A number between 0 and 1, never a negative one. Raising a positive base to a negative power gives its reciprocal, so 10⁻² is 0.01 and e⁻¹ is about 0.368. The result gets smaller as the exponent falls but never reaches zero and never crosses below it, which is the mirror of the fact that logarithms of numbers under 1 are negative. A base-10 antilog only reaches zero once the exponent drops past about −323.6, and that is the arithmetic running out of room rather than the mathematics changing.

Antilog Calculator

Take the antilogarithm of any exponent in any base, and see what that answer is actually worth given how precisely you knew the exponent. An antilog is not a separate function. antilog₍b₎(y) is b raised to y and nothing else. The word is a survivor from printed log tables, where you looked a number...

antilog₍b₎(y) = b^y

It is not a separate function — the word is a leftover from printed log tables. What matters is that going this direction magnifies error: an absolute slip in the exponent becomes a relative slip in the answer, multiplied by the natural log of the base.

THE LOGARITHM

yBASE

y IS KNOWN TO

ANTILOG

2999.85318119079

Taking the logarithm back gives 3.4771 — exactly y again.

CHARACTERISTIC

MANTISSA

0.477100

TO 4 DP, GOOD TO

3.94 s.f.

If that exponent is only good to 4 decimal places, the answer lies between 2999.5078 and 3000.1986. Half a unit in the last decimal you kept is 0.00005, and multiplying by ln(10) turns that into 0.0115% of the answer — about 3.94 significant figures, however many digits the display shows. This is the direction in which precision is lost, and it is why rounding a logarithm before antilogging it costs more than it looks like it should.

Quantity	Value	What it is
antilog	2999.85318119079	10 raised to y
characteristic	3	whole part of y, which sets the magnitude
mantissa	0.477100	fraction of y, which sets the digits
10^characteristic	1000	a clean power of the base
10^mantissa	2.999853181	always between 1 and 10
log back	3.4771	y again, exactly
range	−323.607 to 308.255	exponents with a representable antilog

Why the word exists at all. A table of logarithms printed only the fractional part, because the whole part was free: it just told you where the decimal point went. To multiply two numbers you looked up both logarithms, added them, and then went back through a second table — the table of antilogarithms — to recover the digits. The operation has no life of its own; antilog₍10₎(y) is 10 raised to y and always was. The name survives because chemistry, acoustics and seismology all quote logarithms as their working units, so “take the antilog” remains a sentence people need.

THE SPLIT

Only the right-hand box ever needed a table. The left is a clean power of the base and just places the decimal point, which is why a page of logarithms could print the fraction alone and let you supply the rest.

Live calculation · updates as you type

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HOW TO USE

1
Enter the exponent y and the base. Base 10 is what "antilog" means when nobody says otherwise; e and 2 are a click away in the presets, and any positive base other than 1 works.
2
Set how many decimal places your exponent is actually good to. This is the setting that matters most: the panel then shows the interval the true answer lies in and how many significant figures survive, which is usually fewer than the display suggests.
3
Read the characteristic and mantissa split. The whole part of the exponent is a clean power of the base and only places the decimal point; the fraction supplies the digits. That split is exactly how a table was used and why only the fraction was ever printed.
4
Check the range strip before trusting a large exponent. It marks where your base overflows and underflows, measured by bisection on your own platform, and shows where your exponent sits between them.

REFERENCE FORMULAS

Rule	Formula	What it is for
Definition	antilogᵇ(y) = bʸ	Not a separate function. The word is a leftover from printed log tables.
Common antilog	antilog(y) = 10ʸ	When no base is stated in chemistry or engineering, it is base 10.
Natural antilog	antiln(y) = eʸ	The inverse of ln, written exp(y) in most languages and on most calculators.
Round trip	logᵇ(bʸ) = y	Reported on this page in units in the last place, so you can see it close.
Split	bʸ = b⌊y⌋ × b^(y − ⌊y⌋)	Characteristic times mantissa. The whole part sets the size, the fraction the digits.
Error growth	Δx / x ≈ ln(b) × Δy	An absolute error in the exponent becomes a relative error in the answer.
Digits bought	dp decimals → dp − 0.06 sig figs	For base 10. Four decimals in the logarithm is just under four significant figures.
pH	[H⁺] = 10^(−pH)	The most common everyday antilog, and why the word is still current in chemistry.
Decibels	ratio = 10^(dB/10)	Power ratio from a decibel figure. Use dB/20 for amplitude rather than power.
Richter	energy ∝ 10^(1.5M)	Each whole step of magnitude is about 31.6 times the energy, not ten.

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Last updated: August 3, 2026 · Overflow limits found by bisection for your own platform · every answer taken back through the logarithm and reported in units in the last place