Common and natural logarithms: characteristic, mantissa and the log table

Base 10, in two parts

The common logarithm has base 10 and is written log, with no base shown. So log 10 = 1, log 100 = 2, log 0.1 = −1 and log 0.001 = −3.

A logarithm has two parts. The characteristic is the whole-number part and shows the size. The mantissa is the decimal part and is always positive.

For a number above 1, the characteristic is the digits left of the point minus 1. Below 1, it is −(zeros after the point + 1), written with a bar: 0.0123 gives 2̄.

The mantissa comes from the log table using the first digits, plus the mean difference for the fourth digit. The natural logarithm, ln, uses base e ≈ 2.71828 instead of 10.

log (number) = characteristic + mantissa. The characteristic comes from counting; the mantissa comes from the table.

Key terms

Characteristic

Shows how big or small the number is.

Mantissa

Read from the log table.

Mean difference

Added to the table value.

Natural log (ln)

Growth and decay in higher maths.

Short questions with model answers

1. Q1. Example 9: find the characteristic of log 0.00045 and log 0.54.

   • 0.00045: −(3 + 1) = −4 = 4̄
   • 0.54: −(0 + 1) = −1 = 1̄

   4̄ and 1̄ (check: 4.5 × 10⁻⁴ and 5.4 × 10⁻¹).

2. Q2. Example 10 (iii): find log 0.0036.

   • characteristic = −(2 + 1) = −3
   • mantissa = 0.5563 (row 36, column 0)
   • 3̄.5563 = −3 + 0.5563 = −2.4437

   log 0.0036 = 3̄.5563 = −2.4437.

Common mistakes

• ✗ Writing log 0.0036 as −3.5563.

  ✓ Only the characteristic is negative: 3̄.5563 = −3 + 0.5563 = −2.4437.

• ✗ Taking the characteristic of log 725 as 3.

  ✓ Digits before the point minus 1: 3 − 1 = 2.

• ✗ Stopping at 1271 for log 13.45.

  ✓ Add the mean difference for the 5: 1271 + 16 = 1287.

MCQs

1. 1. The base of the common logarithm is:

   1. (a) 2
   2. (b) e
   3. (c) 10
   4. (d) 5
   Show answer

   (c) Common log: base 10 (Review Exercise 2 iii).

2. 2. If log 2 = 0.3010, then log 200 is:

   1. (a) 1.3010
   2. (b) 0.6010
   3. (c) 2.6010
   4. (d) 2.3010
   Show answer

   (d) 200 = 2 × 10², same mantissa, characteristic 2 (Review Exercise 2 vi).

3. 3. The characteristic of log 0.0567 is:

   1. (a) 2̄
   2. (b) 1̄
   3. (c) 2
   4. (d) 3̄
   Show answer

   (a) One zero after the point: −(1 + 1) = 2̄.

4. 4. log 10,000 =

   1. (a) 3
   2. (b) 4
   3. (c) 5
   4. (d) 1000
   Show answer

   (b) 10⁴ = 10,000 (Review Exercise 2 viii).

5. 5. ln e equals:

   1. (a) 0
   2. (b) e
   3. (c) 1
   4. (d) undefined
   Show answer

   (c) Any base to the power 1 is itself (Do you know?, p. 34).

Quick revision

• Common log has base 10; natural log ln has base e ≈ 2.71828.
• Characteristic: counted from the digits; mantissa: from the table, always positive.
• log 13.45 = 1 + 0.1287 = 1.1287.