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My

AdditionalMathematicsModules

Form 4Topic 5

Haiya manyak sinang punyama...!

(Version 2012)

by

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

edmet-nklpunya.blogspot.com

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3125

5.1 INDICES AND LAWS OF INDICES (2)

IMPORTANT NOTES:

1. For an index numberan, (read as a raise to the power ofn), where a is the base and n is the index.

2. ao

= 1 3. na

1

= a n

, where a 0 4. a n1

=n

a , where a 0, (read as a raise to the nth

root). 5. a nm

=

n ma

6. Laws of Indices

6.1 am x an = am + n 6.2 am an = amn (or)a

a

n

m

=nma 6.3 (am)n = am x n

6.4

m

b

a

=

m

m

b

a6.5 (ab)m = am bm

7. Equation Involving Indices

Solve the problem 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

x =alog

blogm

1. Evaluate each of the following without using a calculator.

(a) 43 (b) 24(c)

273

2

(d)

(e)

3

4

3

(f) 0.50 (g) 31

8

(h) 7 0

2. Simplify and then evaluate each of the following

(a) 32 x 35 (b) 32 34 (c) (52)3 (d) 43 x 24 162

(e) 43 x 45 (f)3

2

8

4 (g) 32

27x 9

3

2

(h) (5

2)2 (125)

Exercise 5.1

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3. Simplify each of the following expressions. (3)

(a) 2 m + 3 x 4 m 32m (b) 3 n + 2 3 n 1 (c) 92k 3 k+ 1 x 27 k

(d) 25n x 42n x 63n (e) 20a3 5a-5(f)

)1m(2

1m2m

9

813

+

+

4. Solve each of the following.

(a) Show that 23w + 1 = 2(8w) (b) Show that 5 n + 1 + 5 n 3(5cn 1) is divisible by 3 or 9.

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

2)2x

(d) Express 3 m 2 + 3 m + 1 2(3 m) in the simplest form.


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(e) Express 9 x + 1 32x + 12(32 x 1) in the simplest form. (f) Show that 7p + 1 + 7p + 2 is a multiple of 8.

5.2 LOGARITHMS AND LAWS OF LOGARITHMS

(4)

IMPORTANT NOTES:

1. To convert an equation in index form to logarithm form and vice versa.

IfN = ax

, then loga N = x.

2. loga 1 = 0, and loga a = 1.

3. loga (negative number) = undefined. Similarly, loga 0 = undefined.

4. Law of Logarithms:

4.1 logaxy = loga x + logay

4.2 logay

x

= logax loga y (or) loga (x y) = logax loga y

4.3 logaxm

= m logax

5. Change of Bases of Logarithms:

5.1 logab =alog

blog

c

c5.2 logab =

alog

blog

b

b=

alogb

1

6. Equations Involving Logarithms:

6.1 Converting the equation of logarithm to index form, i.e.

logaN = 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.

1. Express the following equations to logarithm form or index form.

(a) 32 = 25 (b) 4 =83

2 (c) logxq = p (d) 1 = 100

(e) 3 = log327

1

(f) log3243 = 5 (g) px= 5 (h) log636 = 2 (i) x

3 = 108 (j) 2 = log5 25

Exercise 5.2


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x

64

1log

3 m

2. Determine the value of x in each of the following equations.

(a) log3 81 =x (b) log4x =2

1(c) logx 125 = 3

(d) log2x = 2 (e) =3

1(e) logx

216

1= 3

3. Find the value of each of the following. (5)

(a) log10 100 = (b) log10 39.94 = (c) antilog 1.498 =

(d) log10 35

1

=

(e) antilog 0.3185 = (f) antilog (0.401) =

4. Find the value of each of the following without using a calculator.

(a) log2 32 (b) log3243

1 (c) log9 9

(d) log5 0.2 (e) log749 (f) logm

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

(a) log2 45 (b) log2 6 (c) log2 (3

125)

(d) log2 1.5 (e) log2 0.6 (f) log2 20

6. Simplify each of the following expression to the simplest form.

(a) 2 log2x log2 3x + log2y (b) loga 5x + 3 loga 2y (c) logbx + 3 logbx + logb (y + 1)

(d) log2 4x log2 3y 2 (e) 3 log3x + log3 3y + log3 2x (f) log5 m log5 4m log5 k


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7. Determine the values of the following logarithms.

(a) log2 7 (b) log3 23 (d) log0.5 8.21 (c) log3 5

8. Given that log2 w = p, express the following in terms ofp. (6)

(a) logw 4 = (b) log8 16w2 =

(c) log432

w (d) log w 64

9. Given that logm 3 = xand logm 4 = y. Express the following in terms ofxand/ory.

(a) log36m = (b) log3m12

(c) log3 m16 (d) log34

m

Exercise 5.3


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Exercise 5.4 SPM QUESTIONS ( 2003 2010 )

1. Solve the equation 82x 3 =

4

1

2x[3 marks]

SPM2006/Paper1

(8

)2. Given that log2 xy = 2 + 3 log2 x log2y, express y in terms ofx. [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. [3 marks]SPM2005/Paper1


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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 logm ( 427m

) in terms ofp and r. [4 marks]

SPM2005/Paper1

(9)

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


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10. Given that log2 b =x and log2 c = y, express log4

c

8b

in terms ofx andy. [4 marks] SPM2007/Paper

1

11. Given that 9(3n1) = 27n, find the value ofn. [3 marks] SPM2007/Paper 1

(10)

12. Given that log4x = log2 3, find the value ofx. [3 marks] SPM2008/Paper

1

13. Solve the equation: 162x 3 = 84x [3 marks] SPM2008/Paper

1


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9

8

14. Given that 3n 3x 27n = 243, find the value of n. [3 marks] SPM2009/Paper

1

15. Given that log8p log2 q =0, expressp in terms ofq. [3 marks] SPM2009/Paper 1

(11

)

16. Solve the problem: 3x + 2 = 3x = [3 marks] SPM2010/Paper

1

17. Given log2 3 = a and log2 5 = b, express log8 45 in terms ofa and b. [3 marks] SPM2010/Paper1


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18. Solve the problem: 2 3x = 8 + 2 3x 1 [4 marks] SPM2011/Paper

1

19. Given log2x = h and log2y = k, express log2y

x3

in terms of h and k. [3 marks] SPM2010/Paper

1

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