Logarithm SS1 Mathematics Lesson Note

LOGARITHMS OF NUMBERS TO BASE 10

In general, the logarithm of a number is the power to which the base must be raised to give that number. i.e if y=nx, then x = logny. Thus, logarithms of a number to base ten are the power to which 10 is raised 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 logarithmic form
2. a) 2-6 = 1/64      b) 35 =243     c) 53 = 125     d) 104 = 10,000 

 Solutions

(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

2. Express the following in index form
3. a) Log2(1/8) = -3      (b)  Log10(1/100) = -2      (c) Log464 = 3      (d)  Log5625 = 4    

(e) Log101000 = 3

Solutions

1. a) Log2 (1/8)= -3

Then 2-3 = 1/8

1. b) Log10(1/100) = -2

Then 10-2 = 1/100

1. c) Log464 = 3

Then 43 = 64

1. d) Log5625 = 4

Then 54 = 625

1. e) Log101000 = 3

Then 103 = 1000

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

Examples:

1. Use the tables to find the log of:

37     (b) 3900 to base ten

Solutions

3. a) 37 = 3.7 X 10

=3.7 X 101(standard form)

=100.5682 + 1 X101 (from table)

=101.5682

Hence log1037 = 1.5682

3. b) 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 numbers there are two parts, an integer (whole number) before the decimal point and a fraction after the decimal point which is also called the mantissa. E.g

Log103900 = 3.5991

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

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 a number whose logarithm is given. It is possible to use the 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 use the integer to place the decimal point correctly in the final number

Example:

Find the antilog of the following logarithms:

1. 0.5682
2. 2.7547
3. 5.3914

Solutions

 Log            antilog

3. 5682          3.700
4. 2.7547       568.4
5. 5.3914       246200

Logarithms of numbers less than 1

 No                    Log            Antilog

3. 8320              3.9201            8320
4. 58.24             1.7652            58.24

CLASSWORK 

1. Find the log of (i) 0.009321 (ii) 0.5454
2. Find the antilog of: (iii) 3.3210 (iv) 1.8113 (v) 0.5813 (vi) 3.2212

MULTIPLICATION AND DIVISION OF NUMBERS USING LOGARITHMS TABLES

1. Evaluate the following using tables
2. a) 4627 X 29.3
3. b) 819.8 ÷ 3.905
4. c) 48.63 X 8.53
5. d) 15.39 X 3.52

Solutions

29. a) 4627 X 29.3

 No        Log

4627    3.6653

29.3    1.4669

135600    5.1322

4627 X 29.3 = 135600 (4 s.f)

819. b) 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

       48.63 X 8.53

15. c)    15.39 X 3.52

No        log

48.63        1.6869

8.53        0.9309

Numerator 2.6178        2.6178

15. d)        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

CLASSWORK

Use logarithms tables to calculate

1) 36.12 X 750.9     (2)    3577 x  31.11       (3) 256.5 ÷   6.45   (4) 113.2 X 9.98

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
6. Evaluate using logarithm table 6.28 X 304 and express your answer in the form A X 10n, where A is a number between 1 and 10 and n is an integer.

7. Use the logarithm table to calculate 6354 X 6.243 correctly to 3 s.f