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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 = amn 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) 24 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 34 (c) (52)3 (a) 32 x 35 (d) 4-3 x 45 2 (e) 43 (f) 43 x 24 162 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 x2 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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