Topic 4: Indices and Logarithms Lecture Notes: section 3.1 ...

[Pages:23]Topic 4: Indices and Logarithms

Lecture Notes: section 3.1 Indices section 3.2 Logarithms

Jacques Text Book (edition 4): section 2.3 & 2.4 Indices & Logarithms

INDICES

Any expression written as an is defined as the variable a raised to the power of the number n

n is called a power, an index or an exponent of a

e.g. where n is a positive whole number,

a1 = a a2 = a ? a a3 = a ? a ? a an = a ? a ? a ? a......n times

Indices satisfy the following rules:

1) where n is positive whole number an = a ? a ? a ? a......n times e.g. 23 = 2 ? 2 ? 2 = 8

2) Negative powers.....

1

a-n = a n

1

e.g. a-2 = a2

e.g. where a = 2

2-1 =

1

2 or

2-2 =

1= 2? 2

1 4

3) A Zero power a0 = 1 e.g. 80 = 1

4) A Fractional power

1

an

=n

a

1

e.g. 92 = 2 9 = 9 = 3

1

83

=

3

8

=

2

All indices satisfy the following rules in mathematical applications

Rule 1 am. an = am+n

e.g. 22 . 23 = 25 = 32

Rule 2

a m an

= am - n

e.g.

23 22

= 23-2 = 21 = 2

________________________________

note: if m = n,

a m

then a n

= am ? n = a0 = 1

________________________________

am

note: a - n = am ? (-n) = am+n

________________________________

a-m

1

note: a n = a-m ? n = a m + n

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Rule 3

(am)n = am.n

e.g. (23)2 = 26 = 64

Rule 4

an. bn = (ab)n

e.g. 32 ? 42 = (3?4)2 = 122 = 144

Likewise,

an bn

=

a b

n

e.g.

if b0

62 32

=

6 2 3

=

22

=

4

Simplify the following using the above Rules:

1) b = x1/4 ? x3/4 2) b = x2 ? x3/2 3) b = (x3/4)8

x 2y 3

4) b = x 4 y

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