Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

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

 
WEEK 8       

TOPIC: LOGARITHMS

CONTENT

• Logarithms of Numbers to Base 10.

• Multiplication and Division of Numbers Using Logarithms Tables.

LOGARITHMS OF NUMBERS TO BASE 10

In general the logarithm of a number is the power to which the base must be raised in order to give that number. i.e if y=nx, then x = logny. Thus, logarithms of a number to base ten is the power to which 10 is raised in order to give that number i.e if y =10x, then x =log10y. With this definition log101000 = 3 since 103= 1000 and log10100 = 2 since 102=100.

Examples:

1. Express the following in logarithms form

1. 2-6 = 1/64      b) 35 =243     c) 53 = 125     d) 104 = 10,000 

Solutions

1. (a) 2-6 = 1

               64

=log2 (1/64) = -6

(b)35 = 243

 = log3243 =5

       (c)53 =125

        = log5125 = 3

        (d) 104 = 10,000

         = log1010000 = 4

1. Express the following in index form

1. Log2(1/8) = -3      (b)  Log10(1/100) = -2      (c) Log464 = 3      (d)  Log5625 = 4    (e) Log101000 = 3

Solutions

1. Log2 (1/8)= -3

Then 2-3 = 1/8

1. Log10(1/100) = -2

Then 10-2 = 1/100

1. Log464 = 3

Then 43 = 64

1. Log5625 = 4

Then 54 = 625

1. Log101000 = 3

Then 103 = 1000

Note: Logarithms of numbers to base ten are found with the help of tables

Examples:

Use the tables to find the log of:

1. 37     (b) 3900 to base ten

Solutions

1. 37 = 3.7 X 10

=3.7 X 101(standard form)

=100.5682 + 1 X101 (from table)

=101.5682

Hence log1037 = 1.5682

1. 3900 = 3.9 X1000

=3.9 X 103 (standard form)

=100.5911 X 103 (from table)

=100.5911 + 3

=103.5911

Therefore log103900 = 3.5911

In logarithms any of the number there are two parts, an integer (whole number) before the decimal point and a fraction after the decimal point which is also called mantissa. E.g

Log103900 = 3.5991

            Integer            decimal fraction(mantissa)

The integer part of log103900 is 3 and the decimal part is .5911

In order to obtain the integer part of the logarithm of a number to base ten, count the number of digits to the left hand side of the decimal point and subtract 1. The decimal fraction part of the logarithm of the given number is obtained from the tables.

Examples:

Use the logarithm table to find the logarithms to base ten of:

1. 51.38      2.  840.3      3.  65160

Solutions

1. Log1051.38 = 1.7108
2. Log10840.3 = 2.9244
3. Log1065160 = 4.8140

Antilogarithms table

Antilogarithm is the opposite of logarithms. To find number whole logarithm is given. It is possible to use logarithm table in reverse

However, it’s convenient to use the tables of antilogarithms. When finding an antilogarithm, look up the fractional part only, then used the integer to place correct the decimal point correctly in the final number

Examples:

Find the antilog of the following logarithms:

1. 0.5682
2. 2.7547
3. 5.3914

Solutions

Log            antilog

1. 5682        3.700
2. 2.7547        568.4
3. 5.3914        246200

Logarithms of numbers less than 1

No        Log            Antilog

8320                3.9201            8320

58.24        1.7652            58.24

Evaluation 

1. Find the log of: (i) 0.009321 (ii) 0.5454

1. Find the antilog of: (iii) 3.3210 (iv) 1.8113 (v) 0.5813 (vi) 3.2212

MULTIPLICATION AND DIVISION OF NUMBERS USING LOGARITHMS TABLES

Evaluate the following using tables

1. 4627 X 29.3
2. 819.8 ÷ 3.905
3. 48.63 X 8.53

15.39 X 3.52

Solutions

1. 4627 X 29.3

No        Log

4627    3.6653

29.3    1.4669

    135600    5.1322

4627 X 29.3 = 135600 (4 s.f)

1. 819.8 ÷ 3.905

No        Log

819.8    2.9137

3.905    0.5916

209.9    2.3221

Therefore 819.8 ÷ 3.905 = 209.9

3.       48.63 X 8.53

    15.39 X 3.52

No        log

48.63        1.6869

8.53        0.9309

Numerator 2.6178        2.6178

        15.39        1.1872

        3.52        0.5465

Denominator1.7337    1.7337

        7.658        0.8841

Therefore 48.63 X 8.53 = 7.658

                   15.39 X 3.52

EVALUATION

Use logarithms tables to calculate

1.         36.12 X 750.9 (2)    3577 x  31.11       (3) 256.5 ÷   6.45

             113.2 X 9.98

GENERAL EVALUATION

1. Change the following to logarithms form

1. 25 ½ = 5      b. (0.01)2 = 0.0001

1. Change the following to index form

1. Log31 = 0    b. Log15225 =2

1. Evaluate the following using logarithms tables

1. 69.7 X 44.63

   25.67

1. 17.9 x 3.576 x 98.14

READING ASSIGNMENT

New General mathematics SSS1, page 21, Exercise 1h 1 – 3.

WEEKEND ASSIGNMENT

1. Find the log of 802 to base 10 (use log tables) (a) 2.9042 (b) 3.9040 (c) 8.020 (d)1.9042 
2. Find the number whose logarithm is 2.8321 (a) 6719.2 (b) 679.4 (c) 0.4620 (d) 67.92
3. What is the integer of the log of 0.000352 (a) 4 (b) 3 (c) 4 (d)3 
4. Given that log2(1/64) = m, what is m ? (a) -5 (b) -4 (c) -6 (d) 3
5. Express the log in index form:  log1010000 =4 (a) 103 = 10000 (b) 10-4 = 10000 (c) 104 = 10000 (d) 105 =100000

THEORY 

1. Evaluate using logarithm table 6.28 X 304

           981

And express your answer in the form A X 10n, where A is a number between 1 and 10 and n is an integer.

1. Use logarithm table to calculate 6354 X 6.243  correct to 3 s.f

                        16.76 X 323