﻿ Indices and Logarithms

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## Indices and Logarithms

LESSON OBJECTIVES

#### By the end of the lesson the learner should be able to

: perform the basic operations on bar numbers accurately.

Logarithms are applied in almost every part of life directly and indirectly.

They help us to simply work involving numbers.  Discover how this is applicable!

LESSON INTRODUCTION
Expressing numbers as powers of 10
Numbers can be written such that 10 is the base. Power of 10 is called a logarithm.
E.g. express the following numbers as powers of 10
1000 = 103
100  = 102
10  = 101
1 = 100
0.1 = 10 -1
0.01=10-2

0.001=10-3

The powers 3,2,1,0,-1,-2 and -3 of the numbers 1000, 100, 10, 1, 0.1, 0.01 and 0.001 respectively are known as their logarithms.
The logarithm of 100, written as log 100 = 2 since 100 = 102
The logarithm of a number can be read form logarithm table.
A part of mathematical table

e.g Logarithm of 4.926 is obtained as follows:
In the column headed x read 4.9. then proceed horizontally to the column headed 2 read 0.6920. (Animate the movement of the arrow for 1.0 to 4.9 and them to 0.6920).
In the mean difference columns proceed. Further horizontally and read 5 in the column headed 6.
To get the logarithm add 5 to 0.6920 as shown below:
0.6920
5+
0.6925
Therefore, Log 4.926 = 0.6925
Other examples:
Example 1
Find Log 6.72
Solution
Write the number in standard form i.e. 6.72 = 6.72 x 10o
From the logarithm tables the log of 6.72 = 0.8274
Example 2

1. Find log 32.3

Solution
Write 32.3 in standard form
32.3 =3.23 x 101
Log 32.3 = 1.5092
Note
The logarithm of a number has two parts: the integral part which is called the characteristic and the decimal part which is called the mantissa. Above6.72 was written as 6.72x106
Example 1

Log 6.72 =0.8274

From example 2, 32.3 is written as 3,23 x 101
log 32.3 = 1.5092
Note:
The powe of 10 when a number is written in standard form is the characteristic.
(Highlight in different colours both he characteristic and mantissa. The colour of the power of 10 should match with that of the characteristic. )
Example 3

1. Find log 0.0079

Solution
Write 0.0079 in standard form
0.0079 = 7.9 x 10-3
In this example the power  of 10 is -3. The characteristic is written as 3 with the negative sign on top i.e.  and read as bar 3. Therefore the logarithm of 0.0079=

For numbers that i.e between 0 and 1 the characteristic is always negative and the mantissa is always positive.
Examples 4
Evaluate the following using logarithms
728 x 32.6
Solution
When multiplying numbers using logarithms write them in standard form and read the logarithms from mathematical tables.  Arrange them as shown below

The logarithms are added to get 4.3753 from the antilogarithm tables.
Note finding antilogarithm is the revenue of finding logarithm.

To find the expected product then read the antilogarithm of 4.3753 from the antilogarithm tables.
Note: Finding antilogarithm is the revenue of finding logarithm.
Using antilogarithm tables read the antilog of 0.3753 (the mantissa)
This gives 2.373
The characteristic 4 in 4.3753 above is simply the power of 10 when the product is written in standard form.
Therefore the expected product is
2.373 x 104 = 23730.
Note:
To multiply two numbers we add the logarithms.

Example 5
Evaluate 48.12 ' 6.43 using logarithm tables.

Solution
Arrange the walking as

 Number Standard form log 48.12 6.43 4.812 x 101 6.43 x 10o 1.6823   - 0.88082 0.7741

To divide the two numbers we subtract their logarithms
Then find the antilogarithm of 0.7741 which is 5.944.
Note that the characteristic is 0. Therefore the answer becomes
= 5.944 x 10o = 5.944 x 1
= 5.944
Using logarithm tables

The working is arranged as follows:

 NUMBER STANDARD FORM FORM 2.61 2.61 x 10o 0.4166 61.72 6.172 x 101 1.7904 21.8 2.18 x 01 2.2070 -1.3385  0.8685

In finding the cube rout of a number is simply multiplying the number with a 1/3 or divide it  by 3
In this case 0.8685 is then divided by 3 i.e

Then find the antilog of 0.2895
Which is 1.948
Note
To find the nth root of a number in an expression  We divide the log of b by n.
Lesson objectives

#### By the end of the lesson, you should you able to perform basicoperations on bar numbers accurately.Expressing numbers as powers of 10Numbers can be written such that 10 is the base. Power of 10 is called a logarithm. v/o For example express the following numbers as powers of 10                                                1000    = 103                                                 100     = 102                                                  10       = 101                                                     1      = 100                                                 0.1 == 10 -1                                                                                                    0.01=                                                                                                   0.001=       The powers 3,2,1,0,-1,-2 and -3 of the numbers 1000, 100, 10, 1, 0.1, 0.01 and 0.001 respectively are known as their logarithms. The logarithm of 100, written as log 100 = 2 since 100 = 102 The logarithm of a number can be read form logarithm table. A part of mathematical table     Add Mean differences x 0  1  2  3  4  5  6  7  8  9 1  2  3  4  5  6  7  8  9 1.0 4.9      0.6920                                                            5         e.g. Logarithm of 4.926 is obtained as follows: In the column headed x read 4.9. then proceed horizontally to the column headed 2 read 0.6920. (Animate the movement of the arrow for 1.0 to 4.9 and them to 0.6920). In the mean difference columns proceed. Further horizontally and read 5 in the column headed 6. To get the logarithm add 5 to 0.6920 as shown below:             0.6920                       5+ 0.6925 Therefore, Log 4.926 = 0.6925 v/o Below are other examples Example 1 Find Log 6.72 Solution Write the number in standard form i.e. 6.72 = 6.72 x 10o From the logarithm tables the log of 6.72 = 0.8274 Example 2 Find log 32.3 Solution Write 32.3 in standard form 32.3 =3.23 x 101 Log 32.3 = 1.5092 Note The logarithm of a number has two parts: the integral part which is called the characteristic and the decimal part which is called the mantissa.  In example 1 above 6.72 was written as 6.72x100 v/o therefore, the                        Log 6.72 =0.8274                       From example 2, 32.3 as written as 3.23 x 101   v/o therefore, the   log 32.3 = 1.5092 Note: The power of 10 when a number is written in standard form is the characteristic.        (Highlight in different colours both he characteristic and mantissa. The colour of the power of 10 should match with that of the characteristic. ) Example 3 Find log 0.0079 Solution Write 0.0079 in standard form             0.0079 = 7.9 x 10-3 In this example the power  of 10 is -3. The characteristic is written as 3 with the negative sign on top i.e.  and read as bar 3. Therefore the logarithm of 0.0079=For numbers that i.e. between 0 and 1 the characteristic is always negative and the mantissa is always positive.Roots of numbers using logarithms Example Evaluate  Using logarithm tables SolutionThe working is arranged as follows: NUMBER STANDARD FORM FORM 2.61 2.61 x 10o                                                                                                                                                                         0.4166 61.72 6.172 x 101                                                                                                                                                                      1.7904 21.8 2.18 x 101                                                                                     2.2070 -1.3385  0.8685                                                                                                   In finding the cube root of a number is simply multiplying the number with a 1/3 or divide it  by 3                     0.8685 is then divided by 3 i.e.  0.8685 = 0.2895                                                      3 Then the antilog of 0.2895 = 1.948 x 10o = 1.948 NB. To find the n the root of a number in an expression              n    6               We divide the log of b by n.Logarithm of numbers between 0 and 1 Logarithm of numbers between 0 and 1 have a negative characteristic but the mantissa is always positive. e.g. Log 0.02 = log (2.0 x 10-2) = 2.3010 Such logarithms are known as bar numbers 2.3010 can be written as -2 + 0.3010 Therefore bar is simply a minus sign written on 2 as 2 and read as 'bar 2' as started earlier.Basic operations on logarithm numbers with negative characteristics. (a) Addition Example 1 Evaluate This is re-written and arranged as: -1 + 0.1502 + -2 + 0.1051 -3 + 0.2553 This gives Example 2 Evaluate Solution Rewrite and arrange as -3 +0.4213 + -1 + 0.7055 (Note: -4+1=-3) -4 + 1.1268 = -3 + 0.1268 = 3.1268 (b) Subtraction Example 1Evaluate Written as - 4+ 0.5082 - -2 + 0.4701 -2+0.0381 = Example 2 Evaluate Solution -1+0.6301 - -3+0.8015 It is not possible to subtract 0.8015 from 0.6301, therefore we borrow (substract) 1 from -1(characteristic i.e. -1 ' 1=-2 and then proceed as follows -2+1.6301- -3+0.8015 1 0.8286 (Note that -2 '3 = -2+3 = 3 -2 =1) Example 3 (iii) -1+0.3701 - -3+0.8123We borrow 1 from 1.3701 and rewrite it as 0+ 1.3701 -2+0.8123 2 + 0.5578 = 2.5578 Multiplication Example 1 Evaluate Re-arrange as -2+0.3501 x 3 - 6+1.0503 = -5+0.0503 = Example 2 Evaluate: Arrange as -2+0.1231 x -2 4 + -0.246 Note: In logarithms the mantissa should be always be positive. In case it is negative, it is changes to be positive as shown below. Borrow 1 form 4 target and then proceed as follows: 3 + 1 -0.2462 = This becomes 3 + 0.7538 = 3.7538. which is equal to (d) Division Example 1 Evaluate Solution Rewrite the number as -2+0.1608 Then divide each by 2 as shown below: Example 2 Evaluate The characteristic should be a whole number that is divisible by 2. We think of a possible way of making it divisible by 2. e.g substracy 1 form the characteristic (-1) and add 1 to the mantissa i.e. -1 ' 1=-2 and 1 + 0.560=1.5060. This gives -2 + 1.5060. Divide (-2+1.5060 by 2 as follows:   Addition of bar numbers Example 1 Evaluate This is re-written and arranged as: -1 + 0.1502   + -2 + 0.1051 -3 + 0.2553 This gives Example 2 Evaluate Solution Rewrite and arrange as -3 +0.4213   + -1 + 0.7055          (Note: -4+1=-3)                   -4 + 1.1268 = -3 + 0.1268                   = 3.1268Subtraction of bar numbers  Example 1 Evaluate             Written as - 4+ 0.5082 -                                 -2 + 0.4701                                   -2+0.0381 = Example 2 Evaluate             Solution             -1+0.6301 -             -3+0.8015 It is not possible to subtract 0.8015 from 0.6301, therefore we borrow (substract) 1 from -1(characteristic i.e. -1 ' 1=-2 and then proceed as follows                   -2+1.6301-                   -3+0.8015                   1    0.8286                   (Note that -2 '3 = -2+3 = 3 -2 =1)                   Example 3 (iii)                                                    -1+0.3701 -                         -3+0.8123                   We borrow 1 from 1.3701 and rewrite it as   0+ 1.3701                                                                                                 -2+0.8123                                                  2 + 0.5578 = 2.5578Multiplication of bar numbersExample 1 Evaluate Re-arrange as -2+0.3501                                        x 3     - 6+1.0503 = -5+0.0503 = Example 2 Evaluate: Arrange as      -2+0.1231                                  x -2                                                   4 + -0.246 Note: In logarithms the mantissa should be always be positive. In case it is negative, it is changes to be positive as shown below. Borrow 1 form 4 targer and then proceed as follows: 3 + 1 -0.2462 = This becomes 3 + 0.7538 = 3.7538. which is equal toDivision of bar numbers Example 1 Evaluate Solution Rewrite the number as -2+0.1608       Then divide each by 2 as shown below:                   Example 2       Evaluate       The characteristic should be a whole number that is divisible by 2.  We think of a possible way of making it divisible by 2. e.g substracy 1 form the characteristic (-1) and add 1 to the mantissa i.e. -1 ' 1=-2 and 1 + 0.560=1.5060. This gives -2 + 1.5060. Divide (-2+1.5060 by 2 as follows

Indices and Logarithms

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