Lesson Notes By Weeks and Term - Senior Secondary School 1

LOGARITHMS (contd.)

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SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

  • New General Mathematics SSS 1     M.F. Macrae et al 

 

 

 
WEEK NINE

TOPIC: LOGARITHMS (contd.)

  1. Relationship between Indices and Logarithms.
  2. Calculation Involving Powers and Roots.

Number                    Power of 10

1000                        103

100                        102

10                        101

1                        0

0.1                        10-1

0.01                        10-2

0.001                        10-3

The table above shows that 1000 is to the power 0f 3, thus the logarithms 0f 1000 to base 10 is 3. In this case, the logarithms of a number is the power to which 10 is raised to give that number. Thus a logarithm is another word for power or index.

 Logarithms can be found in other bases apart from base 10

For example, since 32 = 25, then log232 = 5 i.e log of 32 to base 2 is 5. In general, if y = nX, then x =logny

For logarithms to base 10, the following table can be stated:

 

Power (indices)            logarithms (indices)

1000 = 103            log 1000 =3

100 = 102            log 100 =2

10 = 101            log10 = 1

1 = 100                log 1 =0 

 

Thus, a statement written in index form can be changed to a logarithm form and vice versa.

Examples:

  1. Express the following in logarithms form
  1. 2-3 = 1/8
  2. 36 = 729
  3. 43 = 256

Solutions

  1. 2-3 =  1/8

Then, log21/8 = 3

  1. 36 = 729

Then, log3729 = 6

  1. 43 = 256

Then, log4256 = 3

  1. Express the following in index form
  1. Log10  1     = 3

          100

  1. Log264 = 6
  2. Log5(1/125) = -3

 

Solution

  1. Log10     1         =  - 3

           1000

Then 10-3 = 1/1000

  1. Log264 = 6

Then 26 = 64

  1. Log5(1/125) = -3

Then 5-3 = 1/125

 

EVALUATION

  1. Given that log381=m, then 3m = 81. What is m?
  2. Find the value of log2 128
  3. Fill in the blank box in the statement below

log      343 = 3

 

CALCULATION OF POWERS AND ROOTS USINGS LOGARITHM TABLES

When solving problems of powers and roots using logarithms tables, first find the logarithm of the number and then apply the multiplication or division law of indices to the logarithm value i.e multiply the power with the logarithm and divide the logarithm by root

Examples

Evaluate using logarithm table

  1. 252.82                       2. 6√35.81          3.  √26.21

Solutions

  1. 252.82

No                log

252.82                          2.4028 X 2

Antilog= 63920                 =4.8056

Therefore 252.82 = 63920 = 63900

  1. 6√35.81

No                log

6√35.81            1.5540 ÷ 6

Antilog= 1.816        =0.2590

Therefore 6√35.81 = 1.816 

  1. √26.21

No                log

√26.21            1.4185 ÷ 2

5.121            0.7027

Therefore √26.21 = 5.121

 

EVALUATION 

Evaluate using logarithms tables

  1.   3.533           2.   4√400

CALCULATION INVOLVING MULTIPLE DIVISION, POWERS AND ROOTS USING LOGARITHMS

Example

Evaluate the following using logarithms tables correct to 3.s.f

  1. 94100 X 38.2

5.83 X 8.14

  1. 319.63 X 12.282 X 74
  2. 3   218 X 37.2

      95.43

 

  1. 3  38.32 X 38.2   2

    8.637 X 6.285 

 

Solution 

  1. √94100 X 38.2

5.683 X 8.14

No            log

√94100        4.9736 ÷ 2            2.4868

38.2        1.5821              +1.5821

Numerator                4.0689

5.693        0.754 X3          2.2629

8.14        0.9106            +0.9106

denominator            3.1735

numerator        4.0689

denominator          - 3.1735

7.859                0.8954



Therefore 94100 X 38.2 = 7.859 

               5.582 X 8.14

 

  1. 3√19.63 X 12.282 X 74

No                log 

19.63            1.2930            1.2930

12.282            1.089 X2        +    2.1784

74                1.8692            1.8692

3√19.63 X 12.282 X 74                5.3406 ÷ 3

  1.                     1.7802

 

Therefore 3√19.63 X 12.282 X 74 = 60.29 = 60.3 to 3.s.f

 

  1. 3 19.63 X 37.2

      95.43

No                    log

218                    2.338

37.2            +    1.5705

Numerator             -    3.9090   

Deno. 95.43            1.9797

  1. ÷3
  1. 0.6431

 

        3 19.63 X 37.2 = 4.397 =4.40 to 3 s.f

      95.43

 

  1. 3 38.32 X 2.964 2

   8.637 X 6.285







No            log

38.32        1.5834

2.961        0.4719

Numerator        2.0553        2.0553

8.637        0.9364

6.285        0.7983

Denominator    1.7347        1.7347

            0.3206 X 2

            0.6412 ÷ 3

1.636        0.2137

 

3 38.32 X 2.964 2    = 1.636

   8.637 X 6.285  



EVALUATION

Calculate the following

  1. 3 1064

   29.4

  1.  403.9 X 5.78  2

70.62 X 2.931 

 

GENERAL EVALUATION

  1. If log 0.04 = m and 5m = 0.04 find the value of m
  2. Evaluate the following using logarithms table 
  1. (35.61)2 X 5.62

3√143.5

  1. 3 634.6 2

    21.5

 

READING ASSIGNMENT

NGM SSS1,pages 21-22, exercise 1h no 4 - 12. 

 

WEEKEND ASSIGNMENT

  1. Use the table to find the logarithm of 3.7 (a) 0.5682      (b) 1.5682    (c) 2.5682    (d) 3.5682
  2. Evaluate 3√35                              (a)7.047      (b) 7047     (c) 704.7      (d) 0.7047
  3. Write down the integer of log 25.82 (a) 0 (b) 1 (c) 2 (d) 1 
  4. Use table to find the log of 12.34 (a) 12.35 (b) 1.09913 (c) 2.0913 (d) 1.234
  5. Find the number whose logarithms is 2.8321 (a) 679.2 (b) 679.4 (c) 0.4620 (d) 46.2

 

THEORY

  1. Use the logarithms tables to evaluate 

28.612 X 74.23

355.6 X 2.547

  1. 403.93

79.62






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