My Add Maths Modules - Indices And Logarithm

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					Additional Mathematics Form 4
Topic 5:

DELIGHT
(Version 2007)

by NgKL
(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)

5.1

INDICES AND LAWS OF INDICES

2.

IMPORTANT NOTES:
1. For an index number an, (read as a raise to the power of n), where a is the base and n is the index. ao = 1

2. 3.

1 = a n , where a  0 an

4. 5. 6.

an =
m

1

n

a , where a  0, (read as a raise to the nth root).
n

an =

am

Laws of Indices 6.1 am x an = am + n 6.2 am  an = am  n
(or)

6.4 6.5

(ab)m = am bm
a   b  
m

m = am

b

a n a

m

=

amn

6.3 (a ) = am x n 7. Equation Involving Indices Can be solve by; 7.1 Comparing the indices or bases on both sides of the equation; 7.1.1 If am = an, then m = n. 7.1.2 If am = bm, then a = b. 7.2 Applying logarithms on both sides of the equation; ax = bm log ax = log bm x log a = m log b m log b x= log a

m n

Exercise 5.1:
1. Evaluate each of the following without using a calculator. (b) 24

3.

(a) 43

(c) 3 3

2

(d)     




3 4

3

(e) 0.50

(e) 8



1 3

2.

Simplify and then evaluate each of the following. (b) 32  34 (c) (52)3

(a) 32 x 35

(d) 4-3 x 45

2 (e) 43

(f) 43 x 24  162

8

2

1

(g) 27 3 x (-9) 2

(h) (

2 2 ) 5

 (125) 3

2

(i) (18 x 32) 2

3

3.

Simplify each of the following expressions.
n+3

4.

(a) 2

x 4  32
n

n

(b) 3

n+2

3

n-1

(c) 92n  3n + 1 x 27 n

(d) 25n x 42n x 63n

(e) 20a3  5a-5

(f)

3m  2  81x 1 9 2( x 1 )

4.

Solve each of the following.

(a) Show that 7p + 1 + 7p +2 is a multiple of 8.

(b) Show that 5n + 1 + 5n – 3(5n – 1) is divisible by 3 or 9.

(c) Show that 22x + 3  (9x + 1 – 32x) = ( 2 )2x
3

5.2 LOGARITHMS AND LAWS OF LOGARITHMS
IMPORTANT NOTES:
1. To convert an equation in index form to logarithm form and vice versa. If N = ax , then loga N = x. 2. 3. loga 1 = 0, and loga a = 1. loga (negative number) = undefined. Similarly, loga 0 = undefined. Law of Logarithms: 4.1 loga xy = loga x + loga y 4.2 loga x = loga x – loga y
y

5.

4.

(or) loga (x  y) = loga x – loga y

4.2 loga xm = m loga x 5. Change of Bases of Logarithms: 5.1 loga b =

log c b log c a log b b log b a
=

5.2 loga b = 6.

1 log b a

Equations Involving Logarithms: 6.1 Converting the equation of logarithm to index form, i.e. loga N = x, then N = ax 6.2 Express the left hand side, LHS and the right hand side, RHS, as single logarithm of the same base. Then make the comparison, i.e; (i) loga b = loga c, then b = c. (ii) loga m = logb m, then a = b.

Exercise 5.2:
1. Express the following equations to logarithm form or index form. (b) 4 = 8 3
2

6.

(a) 32 = 25

(c) 1 = 100

(d) px = 5

(e) log6 36 = 2

(f) log3 243 = 5

(f) 3 = log3

1 27

(g) logx q = p

2.

Determine the value of x in each of the following equations. (b) log4 x =

(a) log3 81 = x

1 2

(c) log x 125 = 3

(d) log2 x = 2

(e) log

1 64

x =

1 3

(e) log x

1 = 3 216

3.

Find the value of each of the following.

7.

(a) log10 100 =

(b) log10 39.94 =

(c) log10 1

=

(d) antilog 1.498 =

35

(d) antilog 0.3185 =

(e) antilog ( 0.401) =

4.

Find the value of each of the following without using a calculator. (b) log3

(a) log2 32

1 243

(b) log5 0.2

(d) log9

9

5.

Given that log2 3 = 1.585 and log2 5 = 2.32, find the values of the following logarithms. (b) log2 6

(a) log2 45

(c) log2 1.5

(d) log2 (

125 ) 3

6.

Simplify each of the following expression to the simplest form.

8.

(a) 2 log2 x  log2 3x + log2 y

(b) loga 5x + 3 loga 2y

(c) logb x + 3 logb x + logb (y + 1)

(d) log2 4x – log2 3y – 2

7.

Determine the values of the following logarithms.

(a) log2 7

(b) log3 23

(c) log3 5

(d) log0.5 8.21

8.

Given that log2 w = p, express the following in terms of p. (b) log8 16w2 =

9.

(a) log w 4 =

(c) log4

w 32

(d) log

w

64

9.

Given that log m 3 = x and log m 4 = y. Express the following in terms of x and/or y.

(a) log 36 m =

(b) log3m 12

(c) log3 16 m

(d) log3

m 4

Exercise 5.3:
1. Solve each of the following equations. (b) 22x + 3 = 32

10.

(a) 3x + 2 = 81x

(c) 3x . 4x = 125x + 2

(d) 52x =

125 25 x

2.

Solve each of the following equations.

(a) log10 (2x + 7) = log10 21

(b) log5 (x – 5) =

1 log5 125 3

11.

(c) log2 (x – 3) – log2 (x2 – 9) = 0

(d) 2 log3 2 + log3 (4x – 1) = 1 + log3 (x + 8)

(e) logx 18 – logx 2 = 2

(f)

log2 8 + log4 M = 5

Exercise 5.4 – SPM QUESTIONS (2003 – 2007)
1. Solve the equation 82x – 3 =
1 4
x2

12.

[3 marks] SPM2006/Paper1

2. Given that log2 xy = 2 + 3 log2 x – log2 y, express y in terms of x. [4 marks] SPM2006/Paper1

3. Solve the equation 2 + log3 (x – 1) = log3 x.

[3 marks] SPM2006/Paper1

4. Solve the equation 2x + 4  2x + 3 = 1.

13.

[3 marks] SPM2005/Paper1

5. Solve the equation log3 4x – log3 (2x – 1) = 1

[3 marks] SPM2005/Paper1

6. Given that logm 2 = p and logm 3 = r, express log m ( p and r.

27m ) in terms of 4 [4 marks] SPM2005/Paper1

14.

7. Solve the equation 324x = 48x + 6.

[3 marks] SPM2004/Paper1

8. Given that log5 2 = m and log5 7 = p, express log5 4.9 in terms of m and p. [4 marks] SPM2004/Paper1

9. Solve the equation 42x -1 = 7x.

[4 marks] SPM2003/Paper1

15.

10. Given that log2 b = x and log2 c = y, express log4  8b  in terms of x and y.    c  (4 marks) (SPM2007/Paper 1)

11. Given that 9(3n−1) = 27n, find the value of n.

(3 marks) (SPM2007/Paper 1)


				
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