Antilog Table: How to Read and Use It (With Examples)

TL;DR

An antilog table reverses a logarithm: given ( \log x = y ), it recovers ( x = \text{antilog}(y) ). This article shows how to split a logarithm into its characteristic and mantissa, read the row, column, and mean-difference values, place the decimal point, and handle negative characteristics.

What Is an Antilog Table?

An antilog table is a printed reference that finds the number whose logarithm you already know. If ( \log_{10} x = y ), then ( x = \text{antilog}(y) = 10^{y} ). The table does the job of raising 10 to a decimal power without a calculator. It is the same tool students reach for in exams where calculators are not allowed.

The word "antilogarithm" simply means the inverse of a logarithm. Where a log table takes a number and returns its logarithm, the antilog table runs the process backwards.

Quick Reference:

Definition: The antilog of ( y ) is the number ( x ) such that ( \log_{10} x = y ); that is, ( x=10^{y} ).

Symbol/Notation: ( \text{antilog}(y) ) or ( \log^{-1}(y) )

Value depends on: the mantissa (digits) and the characteristic (decimal placement)

Type: Reference table / inverse function of ( \log_{10} )

Used in: logarithmic calculation, slide-rule-era arithmetic, chemistry (pH), physics, exam computation

The Antilog Formula

The antilog is defined by a single relationship:

[ x = \text{antilog}(y) = 10^{y} ]

In words, the antilog of ( y ) is 10 raised to the power of ( y ). It is just the logarithm relationship read backwards: if ( \log_{10} x = y ), then ( x = 10^{y} ).

Because the base is 10, the formula also explains the table's two-part method: the whole-number part of ( y ) fixes the power of 10 (the size of the answer), while the fractional part fixes the leading digits.

The Two Parts of a Logarithm: Characteristic and Mantissa

Every logarithm value splits into two pieces:

• Characteristic — the integer part (the digits before the decimal point). It fixes where the decimal point goes in the answer and can be positive or negative.
• Mantissa — the fractional part (the digits after the decimal point). It fixes which digits the answer contains and must always be kept positive.

For ( \log x = 2.5678 ):

• Characteristic = 2
• Mantissa = 0.5678

The antilog table reads the mantissa to get the digits, and the characteristic tells you how to position the decimal point afterward. Keep these two jobs separate and the whole method stays clean.

How to Read the Antilog Table: The Layout

An antilog table looks like a grid. Reading it uses four digits of the mantissa in three moves:

• Rows are labeled by the first two digits of the mantissa (for example, .57).
• Main columns are labeled 0 through 9 and are selected by the third digit of the mantissa.
• Mean-difference columns sit on the far right, also labeled 0 through 9, and are selected by the fourth digit of the mantissa. You add this small value to the main reading.

One point that trips up almost everyone the first time: the antilog table is entered using the mantissa only. Unlike a log table, you do not locate the row using the leading digit of the original number — because you don't have the original number yet. That is what you are solving for.

Steps to Find the Antilog of a Number

Here is the full procedure. Each step does one thing.

1. Separate the logarithm into its characteristic and mantissa.
2. Read the mantissa's first two digits as the row.
3. Read the mantissa's third digit as the main column; note the four-figure value at that row–column crossing.
4. Read the mantissa's fourth digit in the mean-difference columns; add that value to the reading from step 3.
5. Place the decimal point so that the number of digits before the decimal equals (characteristic + 1).

That last step is the one to memorize as a rule of its own, so let's make it explicit.

Where does the decimal point go?

For a positive characteristic, the answer has (c+1) digits before the decimal point. If c=2, the answer has 3 digits before the point.

Worked Example: Antilog of a Positive Logarithm

Find antilog(2.5678).

Step 1 — Split it:

characteristic=2, mantissa=0.5678

Step 2 — Row from the first two mantissa digits: row .56.

Step 3 — Main column from the third digit 7: the row–column value reads 3690.

Step 4 — Mean difference from the fourth digit 8: add 7, giving 3697.

Step 5 — Place the decimal. Characteristic is 2, so (2+1)=3 digits sit before the point:

[ \text{antilog}(2.5678) = 369.7 ]

Final answer: ( \text{antilog}(2.5678) = 369.7 )

Worked Example: Antilog of a Negative (Bar) Logarithm

Find antilog(3ˉ.2778), where 3ˉ means the characteristic is −3 while the mantissa 0.2778 stays positive.

Step 1 — Characteristic =−3, mantissa =0.2778.

Step 2 to 4 — Read the mantissa:

• Row .27, main column 7 gives 1897.
• Mean-difference digit 8 adds 4.

1897 + 4 = 1901

Step 5 — Place the decimal for a negative characteristic. Here the answer is a small decimal. With characteristic −3, the first significant digit sits in the third place after the point:

[ \text{antilog}(3ˉ.2778) = 0.001901 ]

Final answer: ( \text{antilog}(3ˉ.2778) = 0.001901 )

Common Confusions With Antilog Table

Reference readers land here to separate two ideas that look alike.

Antilog table vs. log table. A log table maps a number to its logarithm; the antilog table maps a logarithm back to a number. They are inverse tools.

Characteristic vs. mantissa. The characteristic only moves the decimal point; the mantissa only sets the digits.

Antilog vs. reciprocal of a log. ( \text{antilog}(y) ) is ( 10^{y} ), not ( 1/\log(y) ).

Where the Antilog Table Comes From

Antilog tables descend directly from history and have historically been used by engineers, navigators, and astronomers to multiply enormous numbers.

Practice Questions

1. Find ( \text{antilog}(1.6934) ).
2. Find ( \text{antilog}(0.4771) ).
3. Find ( \text{antilog}(3.9031) ).
4. Find ( \text{antilog}(2ˉ.7160) ).
5. Find ( \log N=2.3010 ).

Answers

1. 49.344
2. 3.000
3. 8000
4. 0.052
5. N≈200.0

Conclusion

• An antilog table finds ( x ) from ( \log x ): ( \text{antilog}(y) = 10^{y} ).
• Split the logarithm into characteristic (decimal placement) and mantissa (the digits).
• Read the mantissa through row, main column, and mean-difference column; add the mean-difference value.
• Place the decimal using (characteristic + 1) digits before the point.
• For negative logarithms, convert to bar notation so the mantissa stays positive.