🔢 Antilog Calculator: Inverse Log and Antilogarithm for Any Base

What is the Antilog Calculator?

A logarithm answers the question: to what power must I raise the base to get this number? The antilogarithm reverses that. If log base b of y equals x, then the antilog base b of x equals y, and y is found by computing b^x. In short, taking the log and taking the antilog undo each other, so antilog base b of (log base b of y) gives back y. This inverse relationship is why the antilog is sometimes written as log^(-1) or simply as the exponential of x.

The most common case is base 10, the common logarithm. Antilog base 10 of x means 10^x. For example, since log base 10 of 1000 is 3, the antilog base 10 of 3 is 10^3 = 1000. The natural antilog uses base e (about 2.71828), so the antilog of x is e^x, the exponential function that appears throughout growth, decay and compound-interest problems. Base 2 is the binary antilog, 2^x, which shows up in computing, information theory and audio bit-depth math.

Before electronic calculators, antilogs were found with a printed antilog table. Any log value splits into two parts: the characteristic, the integer part before the decimal point, and the mantissa, the decimal part after it. To read a four-figure antilog table you look up the first part of the mantissa down the left column, move across to the next digit, add the mean-difference column, and then place the decimal point using the characteristic. The characteristic tells you the order of magnitude: a characteristic of 2 means the answer has three digits before the decimal point, while a characteristic of 0 means one digit. This calculator does the same job in one step with 10^x, but knowing the table method explains why the digits land where they do.

Because the antilog is just exponentiation, the value x can be any real number, positive, negative or zero. A negative exponent gives a value between 0 and 1 (for a base greater than 1), since b raised to a negative power is one divided by b to the positive power. An exponent of 0 always gives 1, because any nonzero base raised to the power 0 equals 1. The base itself must be greater than 0, since a base of zero or a negative number does not define a well-behaved antilog.

Antilogs are everywhere in science because so many natural scales are logarithmic. The pH scale is defined as pH = -log base 10 of the hydrogen-ion concentration, so recovering that concentration means taking an antilog: concentration = 10^(-pH). The decibel scale, the Richter scale, stellar magnitudes and the spectrophotometry relationship between absorbance and transmittance all compress huge ranges into small log numbers, and every time you need the underlying linear quantity you reverse the log with an antilog. Mastering the antilog therefore unlocks the back half of nearly every log-based formula you will meet.

A quick way to sanity-check any antilog is to take the ordinary log of your answer and confirm it returns x. If antilog base 10 of 2.5 is about 316.23, then log base 10 of 316.23 should come back to 2.5. This round-trip check works for any base as long as the log and the antilog share that base, and it is the fastest way to catch a slipped decimal point or a wrong-base mistake.

When to use it

• Reversing a logarithm in a physics, chemistry or engineering calculation to recover the original quantity.
• Converting a pH value back to hydrogen-ion concentration, where concentration = antilog of (negative pH).
• Working with decibels, Richter magnitudes or other log scales where you need the underlying linear value.
• Recovering a value from a slide-rule or antilog-table problem by separating the characteristic and mantissa.
• Solving exponential growth, decay and compound-interest equations that finish with a base-e antilog.
• Checking homework: confirming that the antilog of a log returns the number you started with.

How to use the Antilog Calculator

1. Enter the value x, the exponent or log value you want to convert back.
2. Enter the base b, or tap a quick button for base 10, base e, or base 2.
3. Read the main result b^x, plus the natural antilog e^x and binary antilog 2^x.
4. Verify the answer by taking the log of the result, which should return your original x.
5. Try a negative or fractional x to see how the antilog behaves across the range.

Formula & method

Worked examples

Common antilog values for base 10 (10^x)

Antilog of x = 1 across common bases

How the characteristic places the decimal point (base 10)

Common mistakes to avoid

• Mixing up the base of the antilog and the log. The antilog must use the same base as the logarithm you are reversing. The antilog base 10 of (log base e of y) does not return y. Match the base, or the round trip fails.
• Assuming antilog means natural exponent. Unless a base is stated, antilog usually means base 10 (10^x), not e^x. In many textbooks "antilog" alone is the common antilog. Check which base your problem expects.
• Thinking a negative x is invalid. The value x can be negative, zero or fractional. A negative exponent gives a small positive result between 0 and 1 for a base above 1, never a negative number or an error.
• Confusing antilog with reciprocal. The antilog of x is b^x, not 1 ÷ log(x) or 1 ÷ x. It is the inverse operation of taking a logarithm, which is exponentiation, not a reciprocal.
• Misplacing the decimal point from the characteristic. When using an antilog table, the mantissa only gives the digits. The characteristic decides where the decimal point goes. A characteristic of 2 means three digits before the point, so 4417 becomes 441.7, not 4.417.
• Forgetting the sign on pH and decibel problems. Because pH = -log[H+], recovering the concentration needs antilog of the negative pH, that is 10^(-pH). Dropping the minus sign gives an answer that is off by many orders of magnitude.

Glossary

Antilogarithm

The inverse of a logarithm. The antilog base b of x is b raised to the power x, written b^x.

Logarithm

The exponent to which a base must be raised to produce a given number. Antilog reverses it.

Base

The number that is raised to a power. Common bases are 10 (common), e (natural) and 2 (binary).

Common antilog

The antilog using base 10, equal to 10^x. Often what "antilog" means with no base stated.

Natural antilog

The antilog using base e (about 2.71828), equal to e^x, the exponential function.

Characteristic

The integer part of a logarithm, before the decimal point. It sets the order of magnitude of the antilog.

Mantissa

The decimal part of a logarithm, after the decimal point. It fixes the significant digits of the antilog.

Exponentiation

Raising a base to a power, b^x. Taking an antilog is exactly this operation.

Frequently asked questions