Log Number: We are going to look into the use of logarithm of numbers in this very Article.

The logarithm of a number (let’s say “N”) to a particular base (let’s say “b”) is that power/index to which “b” MUST be raised to give us “N”.

Good knowledge of standard form will help a student when dealing with logarithm of a number.

*Logarithm of numbers greater than 1*

Logarithm of numbers greater than 1 has both *characteristic* and *Mantissa* , the Characteristic is gotten/known by either Standard form or by inspection while the Mantissa is found by using four figure table.

To find the logarithm of a number to base 10, there are two key definitions and they are crucial. They are CHARACTERISTICS and MANTISSA.

**Characteristics of a logarithm:**This is the integral/whole number part of a logarithm. The characteristics is always one (1) less than the number of digits in the integral part.

### Examples of *Numbers greater than 1 and their Characteristics*

Numbers greater than 1 | Their Characteristics |

10 | 1 |

100 | 2 |

875 | 2 |

1000 | 3 |

34.26 | 1 |

3426 | 3 |

342.6 | 2 |

### Example 1, for logarithm of 10000:

Using standard form, 10000 = 1.0000 x 10⁴

Thus the Characteristic = 4. (this is because the number of digits in the integral/whole number part of the logarithm is five and must be one less than the total digits)

**Mantissa of a logarithm:**This is the decimal part of a logarithm and it is found using four figure tables to four decimal of places.

### From example 1 above: To find the Mantissa of 1.0000 using your four figure table.

Note that the Characteristic of 10000 = 4.

Mantissa: look for 10 under 0 and difference 0

10 under 0 = 0000

Difference 0 = (logarithm of numbers doesn’t have difference zero)

Therefore, Log 10000 = 4.000

When a number is divided or multiplied by a power of 10, the Mantissa will not change but the Characteristics will change.

### Example 2: Find the logarithm of 4387

Solution:**First step** is to know the Characteristic of that log number by either Standard form or by inspection.

By standard form: 4387 = 4.387 x 10³, Characteristic = 3.

By Inspection: From what we discussed above, we said that the Characteristic of a logarithm is always one less than the number of digits in the whole number part, number of digits in 4387 = 4, which means that 3 is the characteristic because it is one lesser than four.

**Second step** is to find the Mantissa of 4.387 by using the four figure table.

43 under 8 = 6415

Difference 7 = 7

Adding the two will give us 6422

Therefore the logarithm of 4387 = 3.642

### Example 3: Find the logarithm of 6.243

Solution:

Characteristic by standard form 6.243 = 6.243 x 10°

Thus the Characteristic = 0.

Look for the Mantissa

62 under 4 = 7954

Difference 3 = 2

Add both = 7956

Therefore Log 6.243 = 0.7956

### Example 4: Find the logarithm of 62.43

Solution:

Characteristic by inspection = 1.

Look for the Mantissa

62 under 4 = 7954

Difference 3 = 2

Adding both = 7956

Thus, Log 62.43 = 1.7956

*Logarithms of Numbers less than 1*

To evaluate the logarithm of numbers less than 1 from the four figure table, it differs only by its characteristic. The Characteristic has a negative sign. We represent it by crossing a Bar over the characteristic.

*Note also that the Characteristic of log number less than 1 depends on the number of zeros counting from the left to the first significant figure.*

If a Characteristic of a log number is -3., it is pronounced Bar 3 and should be written this way below:

The reason why it is written like that is because it affects only the Characteristics while the Mantissas remain unchanged.

*Examples of Numbers less than 1, their Standard forms and Characteristics*

Numbers less than 1 | Their Standard forms | Characteristics |

0.923 | 9.23 x 10^{-1} | -1 |

0.0923 | 9.23 x 10^{-2} | -2 |

0.00923 | 9.23 x 10^{-3} | -3 |

0.000923 | 9.23 x 10^{-4} | -4 |

### Example 5: Find the logarithm of 0.4356

Solution:

The Characteristic either by standard form or by inspection is -1.

For the Mantissa:

43 under 5 = 6385

Difference 6 = 6

Add both = 6391

Students can solve any questions on logarithm of numbers once they understand how to get both the characteristic and the Mantissa of a logarithm.

## QUESTIONS AND SOLUTIONS (log number)

### Using the table, find the logarithms of the following numbers.

#### Q1: 6935

Solution:

Characteristic by standard form 6935 = 6.935 x 10³ = 3.

Mantissa:

69 under 3 = 8407

Difference 5 = 3

Add both = 8410

Thus, Log 6935 = 3.8410

#### Q2: 8361

Solution:

Characteristic = 3.

Mantissa:

83 under 6 and difference 1 = 9223

So, Log 8361 = 3.9223

#### Q3: 709

Solution:

Characteristic = 2.

70 under 9 and difference 0 = 8506

Therefore Log 709 = 2.8506

#### Q4: 376.4

Solution:

The Characteristic = 2.

37 under 6 and difference 4 = 5757

Thus, Log 376.4 = 2.5757

#### Q5: 29.41

Solution:

The Characteristic = 1.

29 under 4 and difference 1 = 4684

Log 29.41 = 1.4684

#### Q6: 63.2

Solution:

Characteristic = 1.

63 under 2 = 8007

:- Log 63.2 = 1.8007

#### Q7: 6.247

Solution:

The Characteristic = 0.

62 under 4 and difference 7 = 7957

Log 6.247 = 0.7957

#### Q8: 0.4269

Solution:

42 under 6 and difference 9 = 6303

#### Q9: 0.0986

Solution:

98 under 6 = 9939

#### Q10: 0.00064

Solution:

64 under 0 = 8062

#### Q11: 26.09

Solution:

The Characteristic = 1.

26 under 0 and difference 9 = 4165

Log 26.09 = 1.4165

#### Q12: 396.5

Solution:

Characteristic = 2.

39 under 6 and difference 5 = 5982

Therefore Log 396.5 = 2.5982

#### Q13: 3.042

Solution:

The Characteristic = 0.

30 under 4 and difference 2 = 4832

Log 3.042 = 0.4832

#### Q14: 9.006

Solution:

Characteristic = 0.

90 under 0 and difference 6 = 9545

Log 9.006 = 0.9545

#### Q15: 0.0604

Solution:

60 under 4 = 7810

#### Q16: 63.24

Solution:

The Characteristic = 1.

63 under 2 and difference 4 = 8010

Therefore Log 63.24 = 1.8010

#### Q17: 6.324

Solution:

The Characteristic = 0.

63 under 2 and difference 4 = 8010

Thus, Log 6.324 = 0.8010

#### Q18: 632.4

Solution:

The Characteristic = 2.

63 under 2 and difference 4 = 8010

Log 632.4 = 2.8010

#### Q19: 6324

Solution:

Characteristic = 3.

63 under 2 and difference 4 = 8010

Log 6324 = 3.801p

#### Q20: 69.35

Solution:

Characteristic = 1.

69 under 3 and difference 5 = 8410

Log 69.35 = 1.8410

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