Reading Antilog Tables: Step-by-step Guide + Practice Quiz

Reading Antilog Tables

A clear, step-by-step guide with worked examples, a tiny demo table, a quick calculator to check your answers, and a practice quiz.

Key ideas

• Antilog: The antilogarithm of a number \(x\) (base 10) is \(10^x\).
• Characteristic: The integer part of a common logarithm. It sets the position of the decimal point via \(10^{\text{characteristic}}\).
• Mantissa: The decimal part (always taken positive). You read its antilog from the table.
• Mean difference: A small correction used for the 4th digit of the mantissa to improve accuracy.

Goal: Find \(N = 10^{x}\). Split \(x\) into characteristic and mantissa, read the antilog of the mantissa from the table, then multiply by \(10^{\text{characteristic}}\).

Step-by-step: How to read the table

1. Separate parts: Write \(x\) as \(x = C + m\), where C is the characteristic (integer) and m is the mantissa (decimal, \(0 \le m < 1\)).
2. Locate row: Use the first two digits of the mantissa (e.g., for \(m=0.3456\), row is 34).
3. Locate main column: Use the 3rd digit of the mantissa (e.g., 5 gives the main entry in column 5).
4. Apply mean difference: Use the mean difference for the 4th digit (e.g., 6) by adding the correction to the main entry.
5. Place the decimal: Multiply the table value (antilog of mantissa) by \(10^{C}\) to get the final number.
6. For negative logs: If \(x\) is negative, still read the mantissa as a positive decimal and multiply by \(10^{C}\) where \(C\) is negative (this shifts the decimal to the left).

Tiny demo antilog table (for the idea)

This miniature shows the structure only. In exams, use the official full table provided. Main entries correspond to the first three mantissa digits; mean differences adjust for the 4th.

Row	Col 0	Col 1	Col 2	Col 3	Col 4	Col 5	Col 6	Col 7	Col 8	Col 9
34	2.188	2.193	2.199	2.204	2.210	2.215	2.220	2.226	2.231	2.237
35	2.240	2.246	2.252	2.258	2.264	2.270	2.276	2.282	2.288	2.294
36	2.300	2.306	2.312	2.318	2.324	2.331	2.337	2.343	2.349	2.356

Mean differences would be a small add-on based on the 4th digit; consult your full table for exact values.

Worked example (positive)

Find: \(N = 10^{2.3456}\)

• Characteristic: \(C = 2\)
• Mantissa: \(m = 0.3456\)
• Read table: Row 34, Col 5 gives main entry; use mean difference for 4th digit 6 to adjust.
• Final: \(N \approx (\text{antilog of }0.3456)\times 10^{2}\)

Check: Use the calculator below to verify \(10^{2.3456}\).

Worked example (negative)

Find: \(N = 10^{-1.2374}\)

• Characteristic: \(C = -1\)
• Mantissa: \(m = 0.2374\)
• Read table: Row 23, Col 7, adjust with mean difference for 4.
• Final: \(N \approx (\text{antilog of }0.2374)\times 10^{-1}\) shifts the decimal left by one place.

Tip: The mantissa is used as a positive decimal; the negative characteristic sets the decimal position.

Instant antilog calculator (to validate your table reading)

Enter \(x\):

Use this only to check your answers. In tests, you must read values from the official table and apply mean differences.

Common mistakes to avoid

• Wrong row/column: Don’t mix up the 2nd and 3rd mantissa digits. Row uses first two digits; main column uses the 3rd.
• Ignoring mean difference: The 4th digit needs the mean difference. Skipping it reduces accuracy.
• Decimal placement: Always multiply by \(10^{C}\). This sets the correct magnitude.
• Negative logs: Don’t take a negative mantissa. Use a positive mantissa and apply the negative characteristic.
• Rounding: Be consistent with the table’s precision and your exam’s required rounding.

Practice quiz: Reading antilog tables

1) To read \(10^{x}\) from the antilog table, you split \(x\) into:

Characteristic and mantissa Integer and fraction without using the table Base and exponent Only the mantissa

2) For mantissa \(m=0.4273\), which are row and main column?

Row 42, Col 7 Row 42, Col 3 Row 24, Col 7 Row 47, Col 2

3) The 4th mantissa digit (e.g., the “3” in 0.4273) is handled by:

Ignoring it entirely Subtracting a fixed 0.001 Adding the mean difference for that digit Doubling the main entry

4) After reading the antilog of the mantissa, you obtain the final number by:

Dividing by \(10^{C}\) Multiplying by \(10^{C}\) Adding \(C\) Subtracting the mantissa

5) For \(x=-1.2385\), which statement is true?

Use mantissa \(-0.2385\) and multiply by \(10^{+1}\) Use mantissa \(0.2385\) and multiply by \(10^{-1}\) Use mantissa \(0.2385\) and multiply by \(10^{+1}\) Ignore the characteristic

Built for students: follow the step-by-step, then verify with the calculator, and finally test yourself with the quiz.