Revision Of Logarithm Of Numbers Greater Than One And Logarithm Of Numbers Less Than One

CONTENT

• Standard forms
• Logarithm of numbers greater than one
• Multiplication and divisions of numbers greater than one using logarithm
• Using logarithm to solve problems with roots and powers (no > 1)
• Logarithm of numbers less than one.
• Multiplication and division of numbers less than one using logarithm
• Roots and powers of numbers less than one using logarithm

STANDARD FORMS

A way of expressing numbers in the form A x 10^x 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 10^x 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

Number                                                 Power of 10

1. 10²
2. 10¹
3. 10⁰
   1. 10^-3
   1. 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 are shifted to the left to be after the first significant figure.

Examples

1. Express in standard form (i) 0.08356    (ii) 832.8 in standard form

Solution

i            0. 08356 =  8.356 x 10^-2

ii             832.8     =   8.328 x 10²

2. Express the following in standard form

(a)        39.32   =          3.932   x 10¹

(b)        4.83     =          4.83     x 10⁰

(c)        0.005321 =      5.321 x 10^-3

WORKING IN STANDARD FORM

Example

Evaluate the following leaving your answer in standard form

• 4.72 x 10³ + 3.648 x 10³

(ii)6.142  x  10⁵ + 7.32 x 10⁴

(iii)             7.113 x 10^-5–  8.13 x 10^-6

solution

i.                  4.72 x 10³ + 3.648 x 10³                      

              = [ 4.72 + 3.648 ] x 10³

=  8.368 x 10 ³  

ii.             =   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 1

LOGARITHM OF NUMBERS LESS THAN ONE

USING LOGARITHM TO EVALUATE PROBLEMS OF MULTIPLICATION, DIVISION, POWERS AND ROOTS WITH NUMBERS LESS THAN ONE

WEEKEND ASSIGNMENT

Use table to find the log of the following:

1.         900                  (a) 3.9542         (b) 1.9542          (c) 2.9542           (d) 0.9542

2.         12.34               (a) 3.0899        (b) 1.089         (c) 2.0913          (d) 1.0913

3.         0.000197         (a) 4.2945        (b) 4.2945       (c) 3.2945          (d) 3.2945

4.         0.8                   (a) 1.9031        (b) 1.9031       (c) 0.9031          (d) 2.9031

5.         Use antilog table to write down the number whose logarithms is 3.8226.

            (a) 0.6646      (b)   0.06646     (c) 0.006646    (d) 66.46