Logarithm SS1 Mathematics Lesson Note

REVISION OF LOGARITHM OF NUMBERS GREATER THAN ONE AND LOGARITHM OF NUMBERS LESS THAN ONE.

STANDARD FORMS

A way of expressing numbers in the form A x 10x where 1< A ( 10 and x is an integer, is said to be a standard form. Numbers are grouped into two. Large and small numbers. Numbers greater than or equal to 1 are called large numbers. In this case, the x, which is the power of 10 is positive. On the other hand, numbers less than 1 are called small numbers. Here, the integer is negative.

Numbers such as 1000 can be converted to its power of ten in the form 10x where x can be termed as the number of times the decimal point is shifted to the front of the first significant figure i.e. 10000 = 10⁴. Others include:

1. 100 = 10²
2. 10 = 10¹
3. 1 = 10⁰
4. 1.01 = 10-³
5. 1.10 = 10-¹

Note: One-tenth; one hundredth, etc are expressed as negative powers of 10 because the decimal point is shifted to the right while that of whole numbers is shifted to the left to be after the first significant figure.

Examples

1. Express in standard form 
2. 0.08356    
3. 832.8 

Solution

1. 0. 08356 =  8.356 x 10-²
2. 832.8 = 8.328 x 10²

2. Express the following in standard form
3. a) 39.32 = 3.932 x 10¹
4. b) 4.83 = 4.83 x 10⁰
5. c) 0.005321  = 5.321 x 10-³

WORKING IN STANDARD FORM

Example

Evaluate the following leaving your answer in standard form

4. 4.72 x 10³+ 3.648 x 10³
5. 6.142  x  10⁵ + 7.32 x 10⁴
6. 7.113 x 10-⁵ –  8.13 x 10-⁶

Solution

4. 4.72 x 10³ + 3.648 x 10³                            = [ 4.72 + 3.648 ] x 10³

=  8.368 x 10³

6. =   6.142 x 10⁵ +  7.32 x 10⁴

      =   6.142 x 10⁵ +  0.732 x 10⁵

=  [6.142 + 0.732 ] x 10⁵

=  6.874 x 10⁵

iii. =   7.113 x 10-⁵  – 8.13 x 10-⁶

       =    7.113 x 10-⁵ – 0.813 x 10-⁵

=  [ 7.113 – 0.813 ] x 10-⁵

=  6.3 x 10-⁵

LOGARITHM OF NUMBERS GREATER THAN ONE

The base ten logarithm of a number is the power to which 10 is raised to give that number e.g.

628000 = 6.28 x10⁵

628000 = 100.7980 x 10⁵

             = 100.7980+ 5

             = 105.7980

Log 628000 = 5.7980

If a number is in its standard form, its power is its integer i.e. the integer of its logarithm e.g. log 7853 has integer 3 because 7853 = 7.853 x 10³

Examples:

Use tables (log) to find the complete logarithm of the following numbers.

1. 80030
2. 8     
3.  135.80

Solution

4. 80030 = 4.9033
5. 8 =  0.9031
6. 13580 = 2.1329

CLASSWORK 

Use the table to find the complete logarithm of the following:

1. 183      
2. 89500     
3.  10.1300
4. 7

MULTIPLICATION AND DIVISION OF NUMBERS GREATER THAN ONE USING LOGARITHM

To multiply and divide numbers using logarithms, first express the number as a logarithm and then apply the addition and subtraction laws of indices to the logarithms. Add the logarithm when multiplying and subtract when dividing.

Examples

Evaluate using logarithm.

1. 4627 x 29.3
2. 8198  ÷  3.905
3. 48.63  x  8.53

      15.39