CHAPTER 5 INDICES AND LOGARITHMS What is Indices

Examples of numbers in index form. 33 (3 cubed or 3 to the power

So , why we use indices? Indices can make large numbers much more manageable,

LAWS OF INDICES Multiplication of indices with same base: am an = a m

Division of indices with same base: am ÷ a n = a m n

Raising an index to a power (am)n = amn bmn = (bm)n EXAMPLE: (b

n (ab) = n n a b EXAMPLE: (xy)3 = x 3 y 3

Other properties of index Zero index: a 0 = 1, a 0 Negative index:

Example Solve (a) 91 – x = 27 (b) 2 p + 1 43

Solution (a) (b) (c) (d) x = -0. 5 p = 11 x =

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CHAPTER 5 INDICES AND LOGARITHMS What is Indices?

Examples of numbers in index form. 33 (3 cubed or 3 to the power of 3) 25 (2 to the power of 5) 3 and 5 are known as indices. 27=33, 3 is a base and 3 is an index 32=25, 2 is a base and 5 is an index

So , why we use indices? Indices can make large numbers much more manageable, as a large number can be reduced to just a base and an index. Eg: 1, 048, 576 = 220

LAWS OF INDICES Multiplication of indices with same base: am an = a m + n bm + n = bm bn Example: x 4 x 3 = x 4 + 3 = x 7 y 4 y 7 = y 4+(-7) = y 3 = 2 x+3 = 2 x 23 = 8(2 x) 3 y – 2 = 3 y 3 2 =

Division of indices with same base: am ÷ a n = a m n bm n = bm ÷ bn Example: = c 9 4 = c 5 3 x-2 =

Raising an index to a power (am)n = amn bmn = (bm)n EXAMPLE: (b 4)3 = b 4 3 = b 12 (32)3 = 32 3 = 36 (2 x)2 = 22 x (2 y+1)3 = 23 y + 3 32 c = (3 c)2

n (ab) = n n a b EXAMPLE: (xy)3 = x 3 y 3 23 33 = 63 (ab)-2 = a-2 b-2

Law 5: EXAMPLE:

Other properties of index Zero index: a 0 = 1, a 0 Negative index: a-n Fractional index:

Law 5: EXAMPLE:

Example Solve (a) 91 – x = 27 (b) 2 p + 1 43 – p = (c) Solve the simultaneous equation 2 x. 42 y = 8 5 x. 25 -y = (d) 4 x+3 – 4 x+2 = 6

Solution (a) (b) (c) (d) x = -0. 5 p = 11 x = -1, y = 1 x = -1. 5