Additional Mathematics

Form 4Topic 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.

2. ao = 1

3. = a n , where a 0

4. = , where a 0, (read as a raise to the nth root).

5. =

6. Laws of Indices

6.1 am x an = am + n 6.4 (ab)m = am bm

6.2 am an = am n 6.5 =

(or) =

6.3 (am)n = 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

x =

Exercise 5.1: 3.

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

(a) 43 (b) 24

(c)

(d) (e) 0.50

(e)

2. Simplify and then evaluate each of the following.

(a) 32 x 35 (b) 32 34 (c) (52)3

(d) 4-3 x 45

(e)

(f) 43 x 24 162

(g) x (-9) (h) ( )2 (125) (i) (18 x 32)

3. Simplify each of the following expressions. 4.

(a) 2n + 3 x 4n 32n (b) 3n + 2 3n - 1

(c) 92n 3n + 1 x 27 n (d) 25n x 42n x 63n

(e) 20a3 5a-5

(f)

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) = ( )2x

5.2 LOGARITHMS AND LAWS OF LOGARITHMS 5.

IMPORTANT NOTES:

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

If N = 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 loga xy = loga x + loga y

4.2 loga = loga x – loga y (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 =

5.2 loga b = =

6. 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: 6.

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

(a) 32 = 25

(b) 4 =

(c) 1 = 100 (d) px = 5

(e) log6 36 = 2 (f) log3 243 = 5

(f) 3 = log3 (g) logx q = p

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

(a) log3 81 = x(b) log4 x =

(c) log x 125 = 3 (d) log2 x = 2

(e) x = (e) log x = 3

3. Find the value of each of the following. 7.

(a) log10 100 = (b) log10 39.94 =

(c) log10 =

(d) antilog 1.498 =

(d) antilog 0.3185 = (e) antilog ( 0.401) =

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

(a) log2 32(b) log3

(b) log5 0.2 (d) log9

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 1.5(d) log2 ( )

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 (d) log0.5 8.21

8. Given that log2 w = p, express the following in terms of p. 9.

(a) log w 4 = (b) log8 16w2 =

(c) log4 (d) 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 (d) log3

Exercise 5.3: 10.

1. Solve each of the following equations.

(a) 3x + 2 = 81x (b) 22x + 3 = 32

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

(d) 52x =

2. Solve each of the following equations.

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

(b) log5 (x – 5) = log5 125

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) 12.

1. Solve the equation 82x – 3 = [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

13.4. Solve the equation 2x + 4 2x + 3 = 1. [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 ( ) in

terms of p and r. [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 in terms of x and y. (4 marks)

(SPM2007/Paper 1)

11. Given that 9(3n−1) = 27n, find the value of n. (3 marks)(SPM2007/Paper 1)