Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

Common Logarithms: Characteristic and Mantissa Explained

Updated 10 October 2026

A common logarithm is a log to base 10. It splits into a whole-number part, the characteristic, and a positive decimal part, the mantissa. Find the characteristic by counting digits or zeros, read the mantissa from the log table, and use the antilog table to turn a log back into a number.

Understand Common Logarithms: Characteristic and Mantissa

A common logarithm is a logarithm to base 10. When no base is written, log x means log₁₀ x. If 10 raised to the power n gives x, then log x = n. For example, log 1000 = 3 because 10³ = 1000.

Most numbers are not exact powers of 10, so their logs are not whole numbers. log 200 lies between 2 and 3, because 200 lies between 100 and 1000. We split such a log into two parts. The whole-number part is the characteristic. The decimal part is the mantissa. The mantissa is always positive and lies from 0 up to but not including 1.

The characteristic tells you the size of the number. The mantissa tells you its digits. The numbers 4.56, 45.6, 456 and 0.0456 all have the same digits, so they all have the same mantissa, 0.6590. Only the characteristic changes, because it records where the decimal point sits.

For a number below 1, the log is negative. We still keep the mantissa positive. So log 0.0456 = -2 + 0.6590, written 2̄.6590 (read as bar 2, point 6590). Only the 2 is negative. The .6590 stays positive.

A log table gives the mantissa for a given set of digits. An antilog table does the reverse: from a mantissa it gives the digits of the number. You then place the decimal point using the characteristic. In the exam the needed log values are usually given in the question, so you mainly need to handle the characteristic, the mantissa and the antilog correctly.

Key formulas to remember

Definition of common log: log x = n means 10ⁿ = x; Base 10 is assumed. x must be positive.
Split of a log: log x = characteristic + mantissa; Characteristic is an integer. Mantissa is always 0 or more and less than 1.
Characteristic for a number 1 or more: Characteristic = (number of digits before the decimal point) - 1; log 4560 has characteristic 3. log 45.6 has characteristic 1. log 4.56 has characteristic 0.
Characteristic for a number between 0 and 1: Characteristic = -(number of zeros immediately after the decimal point + 1); 0.5 has no zeros, so characteristic is -1 (1̄). 0.0456 has one zero, so it is -2 (2̄). 0.00025 has three zeros, so it is -4 (4̄).
Number of digits in a whole number: Number of digits = characteristic of its log + 1; Used for questions such as how many digits are in 2²⁰.
Laws of logarithms: log (mn) = log m + log n; log (m ÷ n) = log m - log n; log mⁿ = n log m; These turn multiplication, division and powers into addition, subtraction and multiplication.
Antilog: If log x = y, then x = antilog y = 10ʸ; Use the mantissa to find the digits. Use the characteristic to place the decimal point.

How to solve Common Logarithms: Characteristic and Mantissa questions

Use this order for any question on characteristic, mantissa, log or antilog.

1Write the number and decide if it is 1 or more, or between 0 and 1.
2For a number 1 or more, count the digits before the decimal point and subtract 1. This is the characteristic.
3For a number between 0 and 1, count the zeros right after the decimal point, add 1, and make it negative. Write it with a bar.
4Take the mantissa from the given log value or log table. It depends only on the digits, not on the decimal point.
5Write the log as characteristic + mantissa. Keep the mantissa positive.
6For calculations, apply the log laws, add or subtract the logs, and convert any negative total back to a negative characteristic with a positive mantissa.
7For the antilog, look up the mantissa to get the digits. Then use the characteristic to move the decimal point: a characteristic n means the first digit sits at 10ⁿ.
8Check the size of your answer. A characteristic of 2 must give a 3-digit whole number part.

Quickest way: Count, copy, shift

When to use it: Use this when the question gives you a log value for the same digits and asks for the log or number with a different decimal position.

Copy the mantissa from the given log. It never changes when only the decimal point moves.
Count the new characteristic from the digits or zeros of the new number.
Join them. For an antilog, copy the digits from the mantissa and shift the point by the characteristic.
Shortcut: moving the decimal point one place right adds 1 to the characteristic. Moving it one place left subtracts 1.
Eliminate options whose mantissa differs from the given one. Often only one option keeps the same decimal part.

Common mistakes in Common Logarithms: Characteristic and Mantissa

Writing log 0.0456 = -1.3410 as characteristic -1 and mantissa -0.3410 is wrong; the mantissa must be positive.
Students see a negative log and make both parts negative, so the mantissa ends up negative.
Fix: Keep the mantissa positive. Rewrite -1.3410 as -2 + 0.6590, which is 2̄.6590. The characteristic is -2 and the mantissa is 0.6590.
Taking the characteristic of 0.0456 as -1 or 1̄ by counting only the first zero.
Students forget that the characteristic is one more than the number of zeros.
Fix: Count zeros after the point (one here) and add 1 to get 2, so the characteristic is 2̄.
Taking the characteristic of 4560 as 4 because it has four digits.
Students use the digit count directly.
Fix: Subtract 1 from the digits before the decimal point. Four digits give characteristic 3.
Dividing the characteristic and mantissa of a bar number separately, for example writing 3̄.4 ÷ 2 as 1̄.2. This is wrong. The correct answer is 2̄.7.
Here -3 ÷ 2 = -1.5, which is not an integer. If you divide the parts separately, you have to force -1.5 into a whole number and halve the mantissa on its own. The result does not equal half of 3̄.4.
Fix: First rewrite 3̄.4 as 4̄ + 1.4, so that the characteristic divides exactly. 3̄.4 = -4 + 1.4. Dividing by 2 gives -2 + 0.7 = 2̄.7. Check: 3̄.4 = -2.6, half of it is -1.3, and -1.3 = 2̄.7.
Placing the decimal point wrongly in the antilog.
Students mix up the rule for positive and negative characteristics.
Fix: For antilog of 2.3979 the answer is 2.5 × 10², which is 250. For antilog of 2̄.3979 the answer is 2.5 × 10⁻², which is 0.025.
Using natural log values (base e) or forgetting that the base is 10.
Both are written as log in some books and calculators.
Fix: In this chapter log always means base 10 unless the question states another base.

Worked examples

Example 1

Given log 2.5 = 0.3979, find the characteristic and mantissa of log 0.00025. Then find the number whose log is 2̄.3979.

Show the solution

The digits of 0.00025 are 25, the same as in 2.5, so the mantissa is 0.3979.
The number 0.00025 has three zeros right after the decimal point.
Characteristic = -(3 + 1) = -4, written 4̄.
So log 0.00025 = 4̄.3979, which equals -4 + 0.3979 = -3.6021.
For the antilog, the mantissa 0.3979 gives the digits 2.5.
The characteristic is -2, so multiply 2.5 by 10⁻². This gives 0.025.

Answer: Characteristic = -4 (4̄), mantissa = 0.3979, so log 0.00025 = 4̄.3979. The number with log 2̄.3979 is 0.025.

Example 2

Given log 2 = 0.3010, find the number of digits in 2²⁰.

Show the solution

Let x = 2²⁰.
log x = 20 × log 2 = 20 × 0.3010 = 6.0200.
The characteristic is 6 and the mantissa is 0.0200.
Number of digits = characteristic + 1 = 6 + 1 = 7.
Check: 2²⁰ = 10,48,576, which has 7 digits.

Answer: 2²⁰ has 7 digits.

Exam tips

Questions are usually asked as MCQs on the characteristic of a given number, the mantissa of a related number, or the antilog of a given value. Do these directly with the count rule and skip lengthy table work.
When a log value is given for one number, use it for related numbers with the same digits. Do not recalculate. Only the characteristic changes.
Check options for the bar notation. Options such as 2̄.6590 and -2.6590 are different values, so read the sign carefully.
For digit-count questions, multiply the power by the given log, take the integer part and add 1.
Because there is no negative marking, always mark an answer. Eliminate options with a wrong characteristic first.

Practice questions from Indices and Logarithms

Common Logarithms: Characteristic and Mantissa: frequently asked questions

What are the characteristic and mantissa of a logarithm?

The characteristic is the whole-number part of a log. The mantissa is the positive decimal part. For log 456 = 2.6590, the characteristic is 2 and the mantissa is 0.6590.

How do you find the characteristic of a number less than 1?

Count the zeros right after the decimal point, add 1, and make the result negative. For 0.0456 there is one zero, so the characteristic is -2, written 2̄. The mantissa remains positive.

How do you find the antilog of a number?

Split the number into characteristic and mantissa. Use the mantissa to find the digits from the antilog table, such as 0.3979 giving 2.5. Then shift the decimal point by the characteristic: antilog 2.3979 is 250 and antilog 2̄.3979 is 0.025.

Why does the mantissa stay the same when the decimal point moves?

Moving the decimal point multiplies or divides the number by a power of 10. This only adds or subtracts a whole number to the log. The decimal part, the mantissa, is not affected.