**Logarithm of Number Less Than 1**

The characteristics of the logs of all numbers equal to or greater than 1are equal to or greater than 0, i.e. are all positive or zero, if m is the characteristics of all log_{10} N where N is greater than or equal to 1 then m is greater than or equal to 0. We talk of negative characteristics when we consider logarithms of numbers which lie between 0 and 1 i.e. all decimals numbers less than 1, eg 0.0314. Here again we bring in the standard form of the number, where the power of 10 gives us the characteristics, thus 0.0314 = 3.14 x 10^{-2}, and so -2 is the characteristics of the log of 0.0314.

Alternatively, we obtain the characteristics by doing these two things

- add 1 to the number of zeros between the decimal point and the first significant figure in the given number and,
- make this value obtained negative.

**Example**

There is only one zero between the decimal point and 3, the first significant figure in the given number 0.0314 \ 1 + 1 = 2 then (b) this two is made negative, i.e. -2 So the characteristics is -2. But we put the minus sign on top of the characteristics only, thus `2 pronounced “bar” 2 not “minus” 2even though they have the same value to show that the minus only refers to the characteristics and not to both the characteristics and mantissa part of the log. If the minus was in front like -2.(…) it will seem as if both the characteristics and mantissa are negative. This negative characteristics is more obvious when the number is expressed in its standard form a x 10^{-n }in which case the characteristics is (-n) i.e. `n.

**Example**

Find the characteristics (negative) and hence the logarithm of the following numbers

- 0.051 2. 0.0084 3. 0.0000765

Number | No of zeros between decimal pt. and 1^{st} signf figure +1 |
Standard form of number | Characteristics | Answers | |

a | 0.051 | One + 1® 2 neg | 5.1 x 10^{-2} |
2 | Log of 0.051 =`2.7076 |

b | 0.0084 | Two + 1® 3 neg | 8.4 x 10^{-3} |
3 | Log of 0.084 = `3.9243 |

c | 0.0000765 | Four + 1® 5 neg | 7.65 x 10^{-5} |
5 | Log of 0.0000765 = `5.8837 |

Note in seeking for the logarithm of any number, first decide what the characteristics is. This can be either positive, zero or negative as shown in the examples. Then after finding the characteristic, look up the mantissa part of the log on the log table. The mantissa is always positive. Using the logarithm tables, makes calculation especially with very little or small numbers, easier and more time saving. Great care should be taken in applying the laws of logarithm. The antilogarithms are always positive.

Note: Since logarithm, the laws of indices are the same as the laws of logarithm.

**Use of Logarithm in Solving Problems**

Squares and Square Roots

**Definitions **Square of 2 means 2 x 2 = 4 and written as 2^{2}. The square root of 4 means the number which when multiplied by itself gives 4 which is = 2 or -2. Square root of 4 is written as Ö4. Again 3^{3}, the square of 3 = 9 and Ö9 = 3 or -3. Hence the square root of a number m is that number n which when squared equals the original number m, i.e. if the square root of m=n, then n^{2} = m. So squares and square roots are so connected as described above, the square root being usually the smaller number. Note that each number has two possible square roots, the positive root and the negative root. This is because – x – = +, + x + = +.

**Example **

Ö4 = +2 or -2 because both (+2) and (-2) = 4 and (-2) x( -2) = 4. Hence all squares are positive numbers. The squares and square roots of numbers can be found by several calculative methods such as

- using logarithm table
- using squares and square root tables
- using factor methods
- using long division method (for square root),
- using long multiplication (for squares)

Example Find (i) the square of 76, (ii) the square root of 400.

(i) Using long Multiplication Method

(76)^{2 }= 76 x 76 =

76
X 76 |

456

532

5776

(ii) Using logs

Log of 76 = 1.8808

Log of (76)^{2} = 1.8808

X 2 = 3.7616

Antilog = 5776

\ (76)^{2} = 5776

Using Square Tables

Look up for 76 under 0, we have 5776.

**Reciprocal**

A reciprocal, or *multiplicative inverse,* is simply one of a pair of numbers that, when multiplied together, equal 1. If you can reduce the number to a fraction, finding the reciprocal is simply a matter of transposing the numerator and the denominator. To find the reciprocal of a whole number, just turn it into a fraction in which the original number is the denominator and the numerator is 1.

For example, the reciprocal of 2/3 is 3/2 (or 1-1/2) , because 2/3 x 3/2 = 1. The reciprocal of 7 is 1/7 because 7 x 1/7 = 1.

Decimal numbers, too, have reciprocals. To find the reciprocal of a decimal number, divide 1 by that number. For instance, to find the reciprocal of 1.25, divide 1 by 1.25:

1 ÷ 1.25 = 0.8

The multiplicative inverse of 1.25, therefore, is 0.8.

Understanding reciprocals can simplify many math problems when you understand that dividing by a number is the same as multiplying by the reciprocal of that number. For example

5 ÷ 1/4

is the same as

5 x 4/1 (which is simply 5 x 4, which of course equals 20)

10 ÷ 1/10

3 ÷ 3/8

**EXERCISES**

- Without using table write down the square roots of 0.0009 A. 0.003 B. 0.6 C. 0.03 D. 0.06
- What is the reciprocal of this expression 8 ÷ 1/5? A. 30 B. 30.625 C. 40 D. 1.6
- Find the reciprocal of this expression 10 ÷ 1/10. A. 1 B. 100 C. 1000 D. 0.01
- Find reciprocal of -5 A. 1/5 B. 5/1 C. -1/5 D. -5/1
- The characteristics of the logs of all numbers equal to or greater than 1are equal to or greater than A. 0 B. 1 C. 10 D. 11

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