Logarithm II – SS1 Mathematics Lesson Note (First Term)

The logarithm of a number less than one has a negative characteristic, written in bar notation, and a positive mantissa.

Behavioural objectives

1. State which part of the logarithm of a number less than one is positive and which part can be negative.
2. Add the characteristic and the mantissa separately.
3. Multiply a logarithm in bar notation by a whole number.
4. Divide a logarithm in bar notation by a whole number.
5. Leave answers in bar notation where necessary.

What are the characteristic and the mantissa of a logarithm?

A logarithm has two parts: the characteristic (the whole-number part) and the mantissa (the decimal part).

• Characteristic: the integral part. It is negative for a number less than one.
• Mantissa: the fractional part. It is always positive.
• Bar notation: a bar above the characteristic shows that only the characteristic is negative.

What does bar notation mean?

The number 2̄.4423 means -2 + 0.4423, which equals -1.5577.

• 2̄.4423: the characteristic is -2 and the mantissa is +0.4423.
• 1̄.7477: the characteristic is -1 and the mantissa is +0.7477.
• Wrong reading: 2̄.4423 is not -2.4423, because the mantissa stays positive.

How do we use logarithms for numbers less than one?

Logarithms are used to evaluate multiplication, division, powers and roots involving numbers less than one.

• Multiplication: add the logarithms of the numbers.
• Division: subtract the logarithms of the numbers.
• Powers and roots: multiply or divide the logarithm by the power or root.

This lesson deals with the operations on bar notation that these problems need.

What are the rules for operating on bar notation?

Treat the characteristic as a directed number and the mantissa as a positive decimal.

1. The mantissa is positive, so it is added in the usual way.
2. The characteristic is positive or negative, so it is operated on as a directed number.
3. For multiplication and division, separate the characteristic from the mantissa before operating.
4. Combine the two parts at the end.
5. If the mantissa is 1 or more, carry the whole number to the characteristic.
6. Write the final answer in bar notation where the characteristic is negative.

Comparison of the two parts

How do we add logarithms in bar notation?

To add logarithms in bar notation, add the mantissas, add the characteristics, then carry any whole number from the mantissa.

Example A: no carrying

Add 2̄.3456 and 1̄.5210.

1. Add the mantissas: 0.3456 + 0.5210 = 0.8666.
2. Add the characteristics: -2 + (-1) = -3.
3. Combine: 3̄.8666.

Example B: with carrying

Add 1̄.7240 and 2̄.6150.

1. Add the mantissas: 0.7240 + 0.6150 = 1.3390.
2. Carry 1 to the characteristics: -1 + (-2) + 1 = -2.
3. Keep the mantissa 0.3390.
4. Combine: 2̄.3390.

How do we multiply a logarithm in bar notation by a whole number?

To multiply a logarithm in bar notation, multiply the characteristic and the mantissa separately, then combine them.

1. Write the logarithm as the sum of the characteristic and the mantissa.
2. Multiply both parts by the whole number.
3. If the mantissa is 1 or more, move the whole number into the characteristic.
4. Write the answer in bar notation.

Worked example 1

Simplify 2̄.4423 × 3, leaving the answer in bar notation.

1. 2̄.4423 × 3 = 3(-2 + 0.4423)
2. = -6 + 1.3269
3. = -5 + 0.3269 (the 1 from the mantissa moves to the characteristic)
4. = 5̄.3269

Worked example 2

Simplify 1̄.6021 × 2.

1. 1̄.6021 × 2 = 2(-1 + 0.6021)
2. = -2 + 1.2042
3. = -1 + 0.2042
4. = 1̄.2042

How do we divide a logarithm in bar notation by a whole number?

To divide a logarithm in bar notation, first rewrite the characteristic so that the divisor goes into it exactly.

1. Choose the nearest negative multiple of the divisor that is below the characteristic.
2. Add the same number to the mantissa, so the value does not change.
3. Divide the characteristic by the divisor.
4. Divide the mantissa by the divisor.
5. Combine and write the answer in bar notation.

Worked example 3

Simplify 2̄.2337 ÷ 7, leaving the answer in bar notation.

1. 2̄.2337 ÷ 7 = (-7 + 5.2337) ÷ 7 (-2 became -7, so 5 was added to the mantissa)
2. = -1 + 0.7477
3. = 1̄.7477

Worked example 4

Simplify 5̄.3200 ÷ 2.

1. 5̄.3200 ÷ 2 = (-6 + 1.3200) ÷ 2 (-5 became -6, so 1 was added to the mantissa)
2. = -3 + 0.6600
3. = 3̄.6600

What mistakes should we avoid?

• Making the mantissa negative: the mantissa is always positive.
• Dividing -2 by 7 directly: this gives a fraction. Adjust the characteristic first.
• Forgetting to carry: a mantissa of 1 or more must give its whole number to the characteristic.
• Changing the value: whatever is subtracted from the characteristic must be added to the mantissa.
• Dropping the bar: write the final answer in bar notation where the characteristic is negative.

Evaluation

1. Which part of the logarithm of a number less than one is always positive?
2. How is the characteristic treated when we add or multiply logarithms in bar notation?
3. Simplify 2̄.4423 × 3 and 2̄.2337 ÷ 7, leaving your answers in bar notation.
4. Add 3̄.4000 and 2̄.7000.

Assignment

Simplify the following, leaving your answers in bar notation.

1. 2̄.3150 × 4
2. 3̄.2160 ÷ 4