INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms [email protected]...

18
indices&logarithmsindices&logarith msindices&logarithmsindices&logarit hmsindices&logarithmsindices&logar ithmsindices&logarithmsindices&log arithmsindices&logarithmsindices&l ogarithmsindices&logarithmsindices &logarithmsindices&logarithmsindic es&logarithmsindices&logarithmsind ices&logarithmsindices&logarithmsin dices&logarithmsindices&logarithmsi ndices&logarithmsindices&logarithm sindices&logarithmsindices&logarith msindices&logarithmsindices&logarit hmsindices&logarithmsindices&logar ithmsindices&logarithmsindices&log arithmsindices&logarithmsindices&l ogarithmsindices&logarithmsindices &logarithmsindices&logarithmsindic es&logarithmsindices&logarithmsind ices&logarithmsindices&logarithmsin INDICES & LOGARITHMS Name ........................................................................................

Transcript of INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms [email protected]...

Page 1: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

indices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsin

INDICES &

LOGARITHMS

Name

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

Page 2: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 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

2

1

4 4 2

4

1

16 [2]

1

532 [2]

(ii) mnn mn

m

aaa

3

2

27 2 23( 27) 3 9

2

38 [4]

3

416 [8]

Notes : Zero Index : 0,10 awherea

Examples :

00 0 0 1

5 1, 2.2 1, ( 3) 1, 12

Page 3: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 3

ACTIVITY 1:

Find the value for each of the following;

(a) 1

264 64 8

(b) 3

1

8

[2]

(c) 4

1

16

[2]

(d) 5

2

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]

Page 4: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 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

Page 5: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 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

Page 6: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 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 a

1 = 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’

Page 7: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 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

[-1]

(d) log8 2

[3]

(e) log2 23

[3]

(f) loga a4

[4]

Page 8: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 8

2.3 Finding logarithm of a numbers by using the Laws of Logarithms

ACTIVITY 5:

1. Evaluate each of the following without using calculator.

(a) log 2 32= 5

2log 2 5

(b) log 3 27

[3]

(c) log 3 1

[ 0]

(d) log 3 9

[2]

(e) log 8 64

[2]

(f) log 2 8

[3]

2. Find the value of

(a) log 2 6 + log 2 12 – log 2 18

2

2

6(12)log

18

log 4

2

(b) log 3 18 + 2log 3 6 – log 3 72

[2]

LAWS OF

LOGARITHMS

loga xy = loga x + loga y

loga xm = m loga x

y

xalog = loga x - loga y

Page 9: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 9

(c) 2log 4 2 - 4log 3+ log 4 12

[2]

(d) log 5 45+ 5 5 5log 100 log 10 log 18

[2]

3.. Given that log 2 3 = 1.585 and log 2 5 = 2.322 . Evaluate each of the following.

(a) log 2 15

[3.907]

(b) log 2 75

[6.229]

(c) log 2 20

[4.322]

(d) log 2 1.5

[0.585]

5. Given that log3 2 = 0.6309 and log3 5 = 1.4650. Evaluate each of the following.

(a) log 3 10

[2.0959]

(b) log 3 18

[2.6309]

(c) log 3 45

[3.4650]

(d) log 3 0.3

[-1.0959]

Page 10: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 10

2.4 Simplifying logarithmic expressions to the simplest form .

ACTIVITY 6

1. Express each of the following in terms of log a , log b and/or log c .

(a) log ab

[log a + log b]

(b) log 3 2a b

[3log a + 2log b]

(c) log 3

2ab

[3log a +6 log b]

(d) log ab

c

[log a + log b-logc]

2. Express each of the following in term of log a x and log a y .

(a) log a xy

[log log ]a ax y

(b) log a x2y

3

[2log 3log ]a ax y

(c) log a 2x

y

[2log log ]a ax y

(d) log a 2 3a x

y

[2 3log log ]a ax y

Page 11: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 11

3. Write each of the following expressions as single logarithm:

(a) lg 3 + lg 25 = lg 3 + lg 5

= lg 15

(b) 3 lg 2 + 2 lg 3– 2 lg 6

[log 2]

(c) log 2 x + log 2 y 2

[2

2log xy ]

(d) lg 6 + 2 lg 4 – lg 8

[log 12]

(e) lg x +2lg y - 1

[

2

log10

xy

]

(f) 3 lg x – 2

1lg y

4 + 2

[

3

2

100log

x

y

]

(g) 2log a x - 1 + loga y

[2

loga

x y

a

]

(h) log 3 x + 2log 3 y – 1

[

2

3log3

xy

]

(i) log b x + log b y + 1

[ logb xyb ]

(j) log a x + log a y – 1

[ loga

xy

a

]

Page 12: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 12

3. CHANGE OF BASE OF LOGARITHMS _________ __________________________________________________________________

3.1 Finding logarithm of a number by changing the base of the logarithm to a

suitable base

The base of logarithms can be changed to other base by using a formula :

ACTIVITY 7: 1. Find the value of each of the following. Give your answer correct to four

significant figures.

(a)

5log 2

(b) 8log3

[1.893]

(c) 3log 4

[1.262]

(d) 5.0log 2

[-1.00]

2. Find the value of each of the following without using calculator..

(a) log 2 9. 3log 8

(b) log 3 7. log 7 2. log 2 3 =

[1]

(c) 4log 16 . log 3 125 =

[8]

(d) log 4 5. log 5 3. log 3 7. log 7 64 =

[3]

a

bb

c

ca

log

loglog

When c = b, then a

bb

b

ba

log

loglog

= ablog

1

If c = b, so loga b =

log 2

log 5

0.3010

0.6990

0.431

3 3

3 3

log 9 log 8

log 2 log 3

2 3 6

Page 13: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 13

3.2 Solving Problems involving the change of base and laws of logarithms.

ACTIVITY 8

1. Given that log 2 3 = 1.585 and log 2 5 = 2.322.Find the value of each of the

following.

(a) log3 15 (b) log 33

5

[2.4650] [-0.4650]

2. Given that log2 a = b. Without using the calculator, express the following in terms

of b:

(a) loga 16 (b) log 16 a

[4

b] [

4

b]

(c) log 4 a

(d) log a 32a

[

2

b]

[5

1b ]

3. If log 3 x = r and log 3 y = s , express each of the following in terms of r and s .

(a) log 3 x2y (b) 3

9log

x

y

[2r+s] [2+r-s]

4. If mx 2log and ny 2log , express each of the following in terms of m and n.

(a) 4log xy (b) 2logx y

[1

[ ]2

m n ]

[2n

m]

Page 14: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 14

4. EQUATION INVOLVING INDICES AND LOGARITHMS

4.1 Solving equations involving indices

METHOD:

1. Comparison of indices or base

(i) If the base are the same , when yx aa , then x = y

(ii) If the index are the same , when xx ba , then a = b

2. Using common logarithm

ACTIVITY 9: 1. Solve the following equations:

(a) 3x = 81

(b) 16x = 8

[3

4]

(c) 8x+1

= 4

[1

3 ]

(d) 9x+1

= 3

[1

2 ]

(e) 9x. 3

x-1 = 81

[5

3]

(f) 2x + 3

- 42x

= 0

[1]

(g) 814 x

[ 3 ]

(h) 3

125

1 x

[5 ]

43 3

4

x

x

Page 15: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 15

2. By using common logarithm(log10), solve the following equations and give your

answer correct to two decimal places.

(a) 2x = 3

(b) 87 x

[-1.07]

(c) 42x+1

= 7

[0.2]

(d) xx 3.2 = 18

[1.61]

(e) 46 25 x

[2.11]

(f) 493.2 xxx

[21.68]

3. By using replacement method, solve each of the following equations:

(a) 122 34 xx

(b) 433 21 xx

[x = -1]

(c) 13 3 9x x

[x = 2]

(d) 3 22 2 12x x

[x = 0]

lg 2 lg3

lg3 1.58

lg 2

x

x

4 32 2 2 2 1

2

16 8 1

8 1

1

8

x x

xlet y

y y

y

y

3

2 ,

12

8

2 2

3

x

x

x

Substitute y

x

Page 16: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 16

4.2 Solving Equations involving logarithms

METHOD:

1. For two logarithms of the same base, if nm aa loglog , then m = n .

2. Convert to index form, if nma log , then m = a n.

ACTIVITY 10:

1. Solve the following equations.

(a) lg x = lg 3 + 2 lg 2 –lg 2

(b) 2 lg 3 + lg (2x) = lg (3x + 1)

[1

15]

(c) lg (4x – 3) = lg (x + 1 ) + lg 3

[ 6 ]

(d) lg (10x + 5) – lg ( x+ 4 ) = lg 2

[3

8]

2. Solve the following equations :

(a) lg 25 + lg x – lg (x – 1) = 2

(b) lg 4 + 2 lg x = 2

[5]

23 2 lg

2

6x

2

25lg 2

1

25 10 ( 1)

25 100 100

75 100

4

3

x

x

x x

x x

x

x

Page 17: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 17

(c) lg x + lg (2x – 1) = 1

[5

2,2

]

(d) 4log58log xx

[2]

(e) 2log54log3 xx

[2]

(f) 5 5 5log (2 5) 2log 6 log 4x

[7]

(g) log2 x2

= 3 + log2 (x + 6)

[-4,12]

(h) 3log)6(log 24 x

[3]

Page 18: INDICES AND LOGARITHMS - Penditamuda's Blog · PDF fileIndices and Logarithms zefry@sas.edu.my 2 CHAPTER 5 : INDICES AND LOGARITHMS 1.1 Finding the value of number given in the form

Indices and Logarithms

[email protected] 18

SPM QUESTIONS

1. SPM 2003

Given that 3loglog 42 VT , express T in terms of V .[4 marks] [T = 8 V ]

2. SPM 2003

Solve the equation xx 74 12 . [4 marks] [x = 1.677]

3. SPM 2004

Solve the equation 684 432 xx . [3 marks] [x = 3]

4. SPM 2004

Given that m2log5 and p7log5 , express 9.4log5 in terms of m and p.

[4 marks] [2p – m -1]

5. SPM 2005

Solve the equation 1)12(log4log 33 xx . [3 marks] [2

3x ]

6. SPM 2005

Solve the equation 122 34 xx . [3 marks] [x = - 3]

7. SPM 2005

Given that pm 2log and rm 3log , express

4

27log

mm in terms of p and r .

[4 marks] [3r – 2p + 1]

8. SPM 2006

Solve the equation 2 3

2

18

4

x

x

. [3 marks] [x =1]

9. SPM 2006

Given that 2 2 2log 2 3log logxy x y , express y in terms of x.

[3 marks] [y=4x]

10. SPM 2006

Solve the equation 3 32 log ( 1) logx x . [3 marks] [1

18

x ]