ALGEBRA LOGS AND INDICES (NON REAL WORLD)

Transcription

1 ALGEBRA LOGS AND INDICES (NON REAL WORLD)

2 Algebra Logs and Indices LCHL New Course 206 Paper Q4 (b) 204S Paper Q2 (b) LCOL New Course 204S Paper Q (a) 204S Paper Q (c) 204S Paper Q (d) 203 Paper Q3 (c) JCHL New Course 207 Paper Q0 206 Paper Q9 (a) 206 Paper Q9 (b) LCHL Old Course 202 Paper Q5 (a) 20 Paper Q5 (b) (i) 20 Paper Q5 (b) (ii) 200 Paper Q5 (a) 2009 Paper Q5 (c) 2008 Paper Q5 (b) (i) 2008 Paper Q5 (b) (ii) 2006 Paper Q2 (c) 2006 Paper Q5 (c) 2005 Paper Q (b) (i) 2005 Paper Q5 (c) 2004 Paper Q (b) (ii) 2004 Paper Q5 (b) (ii) 2003 Paper Q2 (c) 2002 Paper Q5 (a) 200 Paper Q5 (b) (i) 2000 Paper Q5 (c) (i) 2000 Paper Q5 (c) (ii)

3 ALGEBRA LOGS AND INDICES (NON REAL WORLD)

4 206 LCHL Paper Question 4 (b) (i) 5+5 Marks Given log a 2 = p and log a 3 = q, where a > 0, write each of the following in terms of p and q: log a 8 3 log a 8 3 = log a 8 log a 3 = log a 2 3 log a 3 = 3 log a 2 log a 3 = 3p q Rules of Logs log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n (ii) log a 9a 2 6 log a 9a 2 6 = log a 9a 2 log a 6 = log a 3a 2 log a 2 4 = 2 log a 3a 4 log a 2 = 2 log a 3 + log a a 4 log a 2 = 2 log a log a 2 = 2 log a log a 2 = 2q + 2 4p = 4p + 2q + 2

5 204 LCHL Sample Paper Question 2 (b) Given that p = log c x, express log c x + log c cx in terms of p. log c x + log c cx = log c x 2 + log c c + log c x = 2 log c x + log c c + log c x = p + + p 2 Note: x = x 2 Note: log a a = Rules of Logs log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n = 3 2 p +

6 ALGEBRA INDICES

7 204 Sample LCOL Paper Question (a) Write 6 2 and 8 2 without using indices. 6 2 = 6 2 = = 8 = 9 Rules of Indices a p a q = a p+q a p aq = ap q a p q = a pq a 0 = a p = a p a q = q a p a q = q a p = q a p ab p = a p b p a b p = ap b p

8 204 Sample LCOL Paper Question (c) Show that a a 3 a 4 simplifies to a. Rules of Indices a p a q = a p+q a a 3 a 4 = a a 2 a 4 3 = a3 2 a 4 = a9 2 a 4 = a 2 3 a p = ap q aq a p q = a pq a 0 = a p = a p a q = q a p a q = q a p = ab p = a p b p q a p a b p = ap b p

9 204 Sample LCOL Paper Question (d) Solve the equation 49 x = 7 2+x and verify your answer. Rules of Indices 49 x = 7 2+x 7 2 x = 7 2+x 7 2x = 7 2+x 2x = 2 + x 2x x = 2 x = 2 49 x = 7 2+x 49 2 = = = 240 a p a q = a p+q a p = ap q aq a p q = a pq a 0 = a p = a p a q = q a p a q = q a p = q a p ab p = a p b p a b p = ap b p

10 203 LCOL Paper Question 3 (c) Solve the equation 27 2x = 3 x+0. Rules of Indices a p a q = a p+q 27 2x = 3 x x = 3 x+0 3 6x = 3 x+0 6x = x + 0 5x = 0 x = 2 a p = ap q aq a p q = a pq a 0 = a p = a p a q = q a p a q = q a p = q a p ab p = a p b p a b p = ap b p

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12 207 JCHL Paper Question 0 0 Marks Write each of the following in the form 2 n, where n Q. Rules of Indices a p a q = a p+q (a) = 2 8 Rules Applied a p a q = a p+q a p = ap q aq a p q = a pq a 0 = a p = a p (b) = = = 2 3 a p q = a pq a q = q a p a q = q a p = q a p ab p = a p b p (c) 8 8 = 8 2 = = q a = aq 8 = 2 3 a p q = a pq a b p = ap b p

13 206 JCHL Paper Question 9 (a) 0 Marks for (i), (ii), (iii) and (iv) Write each of the following numbers in the form 3 k, where k Q. Rules of Indices a p a q = a p+q (i) 9 9 = 3 2 (iv) = 3 3 a p = ap q aq a p q = a pq a 0 = (ii) = 3 0 = 3 3 a p = a p a q = q a p a q = q a p = q a p (iii) = 27 2 ab p = a p b p a b p = ap b p = = 3 3 2

14 206 JCHL Paper Question 9 (b) 5 Marks Write 2n 4, in the form a n b where a, b Z. Rules of Indices a p a q = a p+q 2n 4 = 2 4 n 4 = 6n 4 a p = ap q aq a p q = a pq a 0 = a p = a p a q = q a p a q = q a p = q a p ab p = a p b p a b p = ap b p

15 ALGEBRA LOGS AND INDICES (NON REAL WORLD)

16 202 LCHL Paper Question 5 (a) Solve for x R: log 4 2x + 6 log 4 x = log 4 2x + 6 x = 4 = 2x + 6 x 4x 4 = 2x + 6 4x 2x = x = 0 x = 5 log 4 2x + 6 log 4 x = Rules of Logs log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n

17 20 LCHL Paper Question 5 (b) (i) Solve the equation: log 2 x log 2 x = 4log 4 2 log 2 x log 2 x = 4log 4 2 x log 2 x = = x x 4 = x x 4x 4 = x 4x x = 4 3x = 4 x = 4 3

18 20 LCHL Paper Question 5 (b) (ii) Solve the equation: 3 2x+ 7 3 x 6 = 0 Give your answer correct to two decimal places. 3 2x+ 7 3 x 6 = 0 3 2x x 6 = 0 3 x x 6 = x x 6 = 0 3 y 2 7 y 6 = 0 3y 2 7y 6 = 0 3y + y 6 = 0 3y + = 0 y 6 = 0 3y = y = 6 For simplicity let y = 3 x y = 3 3 x = 3 3 x = 6 Not a Solution x = log 3 6 x =.63

19 200 LCHL Paper Question 5 (a) Solve the equation: log 2 x + 6 log 2 x + 2 = log 2 x + 6 log 2 x + 2 = log 2 x + 6 x + 2 = 2 = x + 6 x + 2 2x + 4 = x + 6 2x x = 6 4 x = 2

20 2009 LCHL Paper Question 5 (c) Solve the simultaneous equations log 3 x + log 3 y = 2 log 3 2y 3 2log 9 x = log 3 x + log 3 y = 2 log 3 xy = = xy 9 = xy 9 = 2y 3 3 y 27 = 2y 2 3y 2y 2 3y 27 = 0 2y 9 y + 3 = 0 2y 9 = 0 y + 3 = 0 2y = 9 y = 3 y = 9 2 log 3 2y 3 2log 9 x = log 3 2y 3 log 9 x 2 = log 3 2y 3 log 3 x 2 log 3 9 = log 3 2y 3 log 3 x 2 2 = log 3 2y 3 2 log 3 x 2 = log 3 2y 3 log 3 x = 2y 3 log 3 = x 3 = 2y 3 x x = 2y 3 3 x = 2y 3 3 x = x = 2 y = 3 not a solution as y > 0

21 2008 LCHL Paper Question 5 (b) (i) Solve the equation 2 x 2 = 8 2x+9 2 x 2 = 8 2x+9 2 x 2 = 2 3 2x+9 2 x 2 = 2 6x+27 x 2 = 6x + 27 x 2 6x 27 = 0 x 9 x + 3 = 0 x 9 = 0 x + 3 = 0 x = 9 x = 3

22 2008 LCHL Paper Question 5 (b) (ii) Solve the equation log e 2x log e x 2 = 2 log e x + 4 log e 2x log e x 2 = 2 log e x + 4 log e 2x + 3 x 2 = log e x x + 3 x 2 = x x 2 + 3x 4x 6 = x 2 + 8x + 6 x 2 9x 22 = 0 x x + 2 = 0 x = x = 2 x = is the only valid solution.

23 2006 LCHL Paper Question 2 (c) f x = b 2x and g x = b +2x, where b is a positive number. Find, in terms of b, the value of x for which f(x) = g(x). f x = g x b 2x = b +2x b 2x = b b 2x = b 2x (b + ) b + = b2x b 2x = b + log b b 2x = log b b + b + 2xlog b b = log b 2xlog b b = log b b + 2x = log b b + x = 2 log b b + x = log b b + 2 x = log b b +

24 2006 LCHL Paper Question 5 (c) (i) (ii) Given two real numbers a and b, where a > and b >, prove that log b a + log a b 2 Under what condition is log b a + log a b = 2 (i) (i) log b a + log a b 2 log a b + log a b 2 log a b log a b log a b 2 2 log a b + 0 log a b log a b 0 log a b 2 0 True log b a + log a b = 2 True when log a b 2 = 0 log a b = 0 log a b = a = b Logs Rule log b a = log a b Logs Rule a >, b > then log a b > 0

25 2005 LCHL Paper Question (b) (i) Express in the form 2 p q, where p, q Z Given = There are four 2 4 s therefore = = Write 4 as 2 2 = When multiplying numbers of the same base we add the powers = 2 9 p 2 Turn into top heavy fraction to leave in the form 2q

26 2005 LCHL Paper Question 5 (c) (i) (ii) Show that = log log a b b a, where a, b > 0 and a, b. Show that =, where c > 0, c. log 2 c log 3 c log 4 c log r c log r! c

27 2004 LCHL Paper Question (b) (ii) Show that simplifies to a constant. 3 + x p x p 3 + x p x p = 3 + x p x p = 3 + x p + 3 x p + x p = 3 3xp + + xp x p + = 3 + 3xp + x p = 3 + xp + x p = 3

28 2004 LCHL Paper Question 5 (b) (ii) Solve log 4 x log 4 x 2 = 2. Rules of Logs log 4 x log 4 x 2 = 2 x log 4 x 2 = = x x 2 2 = x x 2 2x 4 = x x = 4 log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n

29 2003 LCHL Paper Question 2 (c) (i) Solve for y: 2 2y+ 5 2 y + 2 = y+ 5 2 y + 2 = 0 2 2y y + 2 = y y + 2 = 0 Let x = 2 y 2x 2 5x + 2 = 0 2x x 2 = 0 2x = 0 x 2 = 0 2x = x = 2 x = 2 2 y = 2 y = log 2 2 y = 2 y = 2 y =

30 2002 LCHL Paper Question 5 (a) (i) Find the value of x: 8 2 x = 32 (ii) Find the value of x: log 9 x = x = = 2x log = x 2 = x log 9 x = = x 27 = x OR 8 2x = x = x = x = 5 2 = x

31 200 LCHL Paper Question 5 (b) (i) Solve log 6 x + 5 = 2 log 6 x for x > 0. Rules of Logs log 6 x + 5 = 2 log 6 x log 6 x log 6 x = 2 log 6 x x + 5 = 2 x x + 5 = 6 2 x 2 + 5x = 36 x 2 + 5x 36 = 0 x + 9 x 4 = 0 x = 9 x = 4 x > 0 x = 4 log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n

32 2000 LCHL Paper Question 5 (c) (i) Solve for x 2 log 9 x = 2 + log 9 5x + 8, x > 0 2 log 9 x = 2 + log 9 5x + 8 Rules of Logs log a m + log a n = log a mn log a m log a n = log a m n n log a m = log a m n log n m = log a m log a n log 9 x 2 log 9 5x + 8 = 2 x 2 log 9 5x + 8 = 2 x 2 5x + 8 = 9 2 x 2 5x + 8 = 3 x 2 = 5x + 54 x 2 5x 54 = 0 x 8 x + 3 = 0 x 8 = 0 x + 3 = 0 x = 9 x = 3 Invalid solution

33 2000 LCHL Paper Question 5 (c) (ii) Solve for x 3e x 7 + 2e x = 0. 3e x 7 + 2e x = 0 3e x e x = 0 Let y = e x 3y y = 0 3y 2 7y + 2 = 0 3y y 2 = 0 3y = 0 y 2 = 0 3y = y = 2 y = 3 y = e x e x = 3 x = ln 3 y = e x e x = 2 x = ln 2