**A decimal part**, which is always positive, is called the **mantissa of the logarithm of the real number.Mantissa of logarithm of real number defined below.**

**An integral part** which is** positive** or “**a number greater than 1** and **negative part for a number less than 1**, is called **characteristic of logarithm of number.**

**Observation:**

The logarithm of any number consist of two part, which has been described before.

**(i) 14325= 1.4325×10⁴**

**(II) 0.14567= 1.4567×10⁻⁴**

**(III) (3) 1.4522=1.4522×10 ⁰**

**Characteristic of Logarithm of a real number greater > 1**

The first part of the above table shows that if a number has one digit in the integral part, then the **characteristic is 0; **if its integral part has two digits, then the **characteristic is 1** ; with three digits in the integral part, the** characteristic is 2** and **so on.**

**Observe:**

The **characteristic of the logarithm** of a number greater than 1 is **always 1 less than the number** of digit in the** integral part of the number.**

** Mathematical notation writing method;**

when a number b is written in scientific notation:

**b= a×10 ⁿ **

**where 1 ≤ a ≤ 10,**

the power of 10 will give the characteristic of the log b.

**Example:**

Number |
mathematical notation |
Characteristic of logarithm |

1.02 |
1.02×10 ⁰ |
0 |

99.5 |
9.95×10¹ |
1 |

106 |
1.06×10² |
2 and so on |

**Characteristic of Logarithm of a real number less < 1**

**A number has no zero** immediately after the decimal point, the characteristic is **-1,**

** if it has one zero immediately** after the decimal point, the characteristic is **-2**.

**If it has two zero** immediately after the decimal point, the characteristic is** -3**; and more so on.

**Characteristic less than 1 example.**

**Example:**

Number |
mathematical notation |
Characteristic of logarithm |

0.472 |
4.72×10 ‾¹ |
-1 |

.03 |
3.0×10⁻² |
-2 |

0.00445 |
4.45×10⁻³ |
-3 and so on |

**Characteristic of logarithm symbolically < 1 written as **

we write -3 **symbolically 3‾**

we write -2 **symbolically 2¯**

we write -1 **symbolically 1‾ **

**Mantissa of logarithm of real number in table.**

**Fined the mantissa of logarithm of 43.254**

rounding off 43.254

we consider only** four significant digit 4325.**

**(1)** we see **rows corresponding to 43** in log **table.**

**(2) see horizontally** till we reach the column 2.the** number at the intersection is 6355.**

(3) again proceeding horizontally till the mean difference column corresponding to** 5 intersect this row.** We get the number 5 at the intersection.

(4) Adding the two number** 0.6355 and 5.**

We get** 0.6360** as the **mantissa of the logarithm.**

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