CHAPTER 5 INDICES AND LOGARITHMS What is Indices?

Examples of numbers in index form. 3 3 (3 cubed or 3 to the power of 3) 2 5 (2 to the power of 5) 3 and 5 are known as indices. 27=3 3, 3 is a base and 3 is an index 32=2 5, 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 = 2 20

LAWS OF INDICES Multiplication of indices with same base: a m  a n = a m + n b m + n = b m  b n Example: x 4 x 3 = x = x 7 y 4 y  7 = y 4+(-7) = y  3 = 2 x+3 = 2 x  2 3 = 8(2 x ) 3 y – 2 = 3 y  3  2 =

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

Raising an index to a power (a m ) n = a mn b mn = (b m ) n EXAMPLE: (b 4 ) 3 = b 4  3 = b 12 (3 2 ) 3 = 3 2  3 = 3 6 (2 x ) 2 = 2 2x (2 y+1 ) 3 = 2 3y c = (3 c ) 2

(ab) n = a n  b n EXAMPLE: (xy) 3 = x 3  y  3 3 = 6 3 (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) 9 1 – x = 27 (b) 2 p + 1  4 3 – p = (c) Solve the simultaneous equation 2 x.4 2y = 8 5 x.25 -y = (d) 4 x+3 – 4 x+2 = 6

Solution (a) x = -0.5 (b) p = 11 (c) x = -1, y = 1 (d) x = -1.5