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

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 relationship behind the whole table deserves a line of its own.

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 $y$. 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. That split is exactly what you read next.

Before you read a single value, you need to understand the two parts of any logarithm.

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 labelled by the first two digits of the mantissa (for example, .57).

• Main columns are labelled 0 through 9 and are selected by the third digit of the mantissa.

• Mean-difference columns sit on the far right, also labelled 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 memorise as a rule of its own, so let's make it explicit.

Where does the decimal point go?

For a positive characteristic $c$, the answer has $(c + 1)$ digits before the decimal point. If $c = 2$, the answer has $3$ digits before the point; if $c = 0$, it has $1$ digit before the point.

Worked Example: Antilog of a Positive Logarithm

Find $\text{antilog}(2.5678)$.

Step 1 — Split it:

$$\text{characteristic} = 2, \quad \text{mantissa} = 0.5678$$

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 $7$.

$$3690 + 7 = 3697$$

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: $\text{antilog}(2.5678) = 369.7$

You can check the logic: $10^{2.5678} \approx 369.7$, which sits between $10^{2} = 100$ and $10^{3} = 1000$, exactly where a characteristic of $2$ says it should.

Worked Example: Antilog of a Negative (Bar) Logarithm

Negative logarithms are where readers slow down, so walk through this one carefully.

Find $\text{antilog}(\bar{3}.2778)$, where $\bar{3}$ (read "bar three") means the characteristic is $-3$ while the mantissa $0.2778$ stays positive.

The bar notation exists precisely so the mantissa can remain positive. A value written $\bar{3}.2778$ means:

$$-3 + 0.2778 = -2.7222$$

Step 1 — Characteristic $= -3$, mantissa $= 0.2778$ (already positive, which is what we need).

Step 2 to 4 — Read the mantissa $0.2778$ through the table:

• 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:

$$\text{antilog}(\bar{3}.2778) = 0.001901$$

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

Notice the antilog of a negative logarithm is still a positive number — just a small one. An antilog is $10^{y}$, and $10^{y}$ is positive for every real $y$.

Worked Example: Using Antilogs to Finish a Calculation

Antilog tables earn their keep at the last step of a log calculation. Suppose you used logarithms to compute $\log N = 1.8451$ while evaluating a product.

Read the mantissa $0.8451$:

• Row .84, column 5 gives $7000$.

• Mean-difference digit 1 adds $1$, giving $7001$.

Characteristic $1$ means $(1 + 1) = 2$ digits before the point:

$$N = \text{antilog}(1.8451) = 70.01$$

Final answer: $N \approx 70.01$

The logarithm turned a hard multiplication into an addition; the antilog table turned the result back into an ordinary number.

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. A frequent error is entering the antilog table by the leading digit of some number, as you would a log table — but the antilog table is entered by the mantissa alone.

Characteristic vs. mantissa. The characteristic only moves the decimal point; the mantissa only sets the digits. Change the characteristic and the digits stay the same — only the size changes. $\text{antilog}(2.5678) = 369.7$ and $\text{antilog}(1.5678) = 36.97$ share the digits $3697$ because they share a mantissa.

Antilog vs. reciprocal of a log. $\text{antilog}(y)$ is $10^{y}$, not $1 / \log(y)$. The "anti-" means inverse operation, not inverse number.

Where the Antilog Table Comes From

Antilog tables descend directly from John Napier's logarithms (1614) and Henry Briggs's base-10 tables that followed. For three centuries — through the slide-rule era — engineers, navigators, and astronomers multiplied enormous numbers by adding logarithms and then reading an antilog to recover the product. The tables are still printed at the back of exam-board mathematics handbooks because they teach the structure of base-10 numbers, not just an answer.

What Are the Most Common Mistakes With Antilog Tables?

Mistake 1: Reading the antilog table with the whole logarithm

Where it slips in: Right at the start, when a reader carries the characteristic into the table lookup.

Don't do this: Trying to find a row for 2.56 — the characteristic 2 has no place in the grid.

The correct way: Read the mantissa only (.5678) through the table. The characteristic is used once, at the very end, to place the decimal point. Students who have just learned log tables most often reach for the leading digit of a number out of habit — pausing to ask "am I reading digits or placing a decimal?" fixes it.

Mistake 2: Leaving the mantissa negative

Where it slips in: When the logarithm itself is negative, such as $-2.7222$.

Don't do this: Trying to look up a mantissa of $-0.7222$. The table has no negative mantissa rows.

The correct way: Convert to bar form so the mantissa is positive. Write $-2.7222 = -3 + 0.2778 = \bar{3}.2778$, then read $0.2778$ as the mantissa. The habit of always making the mantissa positive first prevents a whole class of errors.

Mistake 3: Miscounting digits before the decimal

Where it slips in: Step 5, placing the decimal point.

Don't do this: Writing $\text{antilog}(2.5678)$ as $36.97$ or $3697$, off by one place either way.

The correct way: Use (characteristic $+ 1$) digits before the point. Characteristic $2$ means $3$ digits: $369.7$. This is the same digit-counting error that once cost real accuracy in the slide-rule era, when a misplaced decimal in a navigation table could put a ship's fix off by an order of magnitude. The digits were right, the scale was wrong. Read more about how logarithms compress scale to see why the decimal place carries so much weight.

Practice Questions

Work these with a printed antilog table, then check the answers below.

1. Find $\text{antilog}(1.6934)$.

2. Find $\text{antilog}(0.4771)$.

3. Find $\text{antilog}(3.9031)$.

4. Find $\text{antilog}(\bar{2}.7160)$.

5. A logarithm calculation ends with $\log N = 2.3010$. Find $N$.

Answers

1. $49.34$

2. $3.000$

3. $8000$

4. $0.05200$

5. $N \approx 200.0$

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 positive characteristics.

• For negative logarithms, convert to bar notation so the mantissa stays positive; the antilog is still positive.

To build fluency with logarithms and antilogs alongside a teacher, explore Bhanzu's algebra tutor sessions or algebra classes, and for exam-level work a high school math tutor can pace the practice.

Read More

• Logarithm rules: product, quotient, and power

• How to read and use a log table

• Properties of logarithms — laws and formulas

• Logarithmic functions: graph and properties