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Transcript of INDICES AND LOGARITHMS - Penditamuda's Blog and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES...

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    INDICES &

    LOGARITHMS

    Name

    ........................................................................................

  • Indices and Logarithms

    zefry@sas.edu.my 2

    CHAPTER 5 : INDICES AND LOGARITHMS

    1.1 Finding the value of number given in the form of :-

    Type of indices In General Examples

    (a) Integer indices (i) positive indices

    aaaaan .....

    n factors

    a = base(non zero number)

    n = index(positive integer)

    53 3 3 3 3 3

    2( 4)

    3)2.0(

    31

    5

    (ii) negative indices

    n

    n

    aa

    1

    12 =1

    2

    23

    2

    4

    (b) Fractional indices (i) nn aa

    1

    n = positive integer

    a 0

    21

    4 4 2

    41

    16 [2]

    1

    532 [2]

    (ii) mnn mnm

    aaa

    32

    27 2 23( 27) 3 9

    2

    38 [4]

    3

    416 [8]

    Notes : Zero Index : 0,10 awherea

    Examples :

    00 0 0 15 1, 2.2 1, ( 3) 1, 1

    2

  • Indices and Logarithms

    zefry@sas.edu.my 3

    ACTIVITY 1:

    Find the value for each of the following;

    (a) 1

    264 64 8

    (b) 31

    8

    [2]

    (c) 41

    16

    [2]

    (d) 52

    32

    [4]

    (e) 2

    327

    [9]

    (f) 1

    225

    [1

    2]

    (g)

    1

    31

    8

    [2]

    (h)

    21

    4

    [16]

    (i)

    11

    32

    [32]

    (j)

    21

    3

    [9]

    EXERCISE 1

    1. Evaluate the following:

    (a) 3

    24

    [8]

    (b) (16)1

    4

    [2]

    (c)

    11

    25

    [25]

    (d)

    1

    532

    81

    [2

    3]

    (e) 1 2(5 )

    [1

    25]

    (f) 1

    24

    [1

    2]

    2. Write in index form

    (a) 1

    p

    [p-1

    ]

    (b) 3

    1

    q

    [q-3

    ]

    (c)

    1

    1

    p

    [p1]

  • Indices and Logarithms

    zefry@sas.edu.my 4

    ACTIVITY 2:

    1. Simplify each of the following:

    (a) 52 aa

    (b) 2 35 5n n

    (c) 6

    2

    x

    x

    (d) 34 2 2n n n

    (e) 1

    6 4 2p q

    (f) 1

    4 12 4a b

    (g) 1

    8 2 281p q

    (h) 416 2n

    (i) 2 33 6 2

    (j) 1 1

    5 332 125

    (k) 1 12 4 2n n

    (l) 2 3

    2 4

    n na a

    a a

    LAWS OF INDICES

    nmnm aaa nmnm aaa mnnm aa )(

    mmm baab )( n

    nn

    b

    a

    b

    a

    2 5

    7

    a

    a

    5[5 ]n

    [1] 4[ ]x

    3 2[ ]p q 3[ ]ab

    4 1[9 ]p q 8[2 ]n

    5 4[3 2 ] 2[ ]5

    [4] 5 6[ ]na

  • Indices and Logarithms

    zefry@sas.edu.my 5

    2. Prove that

    (a) 11 4344 nnn is divisible by 17

    for all positive integers of n .

    17(4n-1

    ) is a multiple of 17 and hence,

    17(4n-1

    ) is divisible by 17.

    (b) 21 555 nnn is divisible by 31 for

    all positive integers of n.

    [31(5n) is a multiple of 31 and hence, 31(5

    n) is

    divisible by 31].

    (c) 21 333 nnn is divisible by 13 for

    all positive integers of n.

    [13(3n) is a multiple of 13 and hence, 13(3

    n) is

    divisible by 13].

    (d) 2 32 2p p is divisible by 12 for all

    positive integers of p.

    [12(2p) is a multiple of 12 and hence, 12(2

    p) is

    divisible by 12].

    1

    1

    1

    44 4 4 3

    4

    34 (4 1 )

    4

    174

    4

    17(4 )

    nn n

    n

    n

    n

  • Indices and Logarithms

    zefry@sas.edu.my 6

    2. LOGARITHMS AND THE LAW OF LOGARITHMS.

    _____________________________________________________________________

    2.1 Express equation in index form to logarithm form and vice versa

    Definition of logarithm

    If a is a positive number and a 1 , then

    `

    (INDEX FORM) (LOGARITHM FORM)

    N = Number

    a = base

    x = index

    We can use this relation to convert from index form to logarithm form or vice versa.

    ACTIVITY 3:

    1. Convert each of the following from index form to logarithm form:

    INDEX FORM LOGARITHM FORM

    (a) 43

    = 64

    4log 64 3

    (b) 34

    = 81

    (c) 2-3

    = 1

    8

    (d)10-2

    = 0.01

    (e) 1

    3 = 13

    2. Convert each of the following from logarithm form to index form:

    LOGARITHM FORM INDEX FORM

    (a) log7 49 = 2

    249 7

    (b) log3 27 = 3

    (c) log9 3 = 1

    2

    (d) log10 100 = 2

    (e) log5 1

    16 = - 4

    Notes! Since a1 = a then loga a = 1

    Since a0 = 1 then loga 1 = 0

    xaN xNa log

    Loga N is read as logarithm of N

    to the base a

  • Indices and Logarithms

    zefry@sas.edu.my 7

    3. Find the value of x .

    a) log2 x = 1

    12 2x

    b) log10 x = -3

    [0.001]

    c) log3 x = 4

    [81]

    d) loga x = 0

    [1]

    2.2 Finding logarithm of a number

    Logarithm to the base of 10 is known as the common logarithm. The value of

    common logarithms can be easily obtained from a scientific calculator .

    In common logarithm, if log10 N = x , then , antilog N = 10x. [lg N = log10N]

    ACTIVITY 4

    1. Use a calculator to evaluate each of the following;

    (a) log10 16 = 1.2041

    (b) log10 0.025 =

    [-1.6021]

    (c) log10 2

    3

    =

    [-0.1761]

    (d) log10 52 =

    [1.3979]

    (e) antilog 0.1383 =

    [1.3750]

    (f) antilog (- 0.729) =

    [0.1866]

    (g) antilog 1.1383 =

    [13.7450]

    (h) 10- 2

    =

    [0.01]

    2. Find the value of the following logarithms.

    (a) log4 16=2

    4log 4 2 (b) log3 27

    [3]

    (c) log2 1

    2