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![5 Indices & Logarithms 12
Enrichment Exercises (SPM Format Questions)
EXERCISE EXCERCISE
L1 Given log3 x = m and log2 x = n.
Find logx 24 in terms of m and n.
[SPM 2001] [4]
(Ans : 3/n + 1/m )
L2. Given log3 x = p and log2 x = q.
Find logx 36 in terms of m and n.
[4]
(Ans: 2/p + 2/q )
L3. Given log3 x = p and log9 y = q.
Find log3 xy2
in terms of p and q.
[SPM 1998] [4]
(Ans: p + 4q )
L4. Given log3 x = p and log9 y = q.
Find log3 x2
y3
in terms of p and q.
[4]
(Ans: 2p + 6q )
L5 Given log5 2 = m and log5 7 = p, express
log5 4.9 in terms of m and p. [4]
[SPM 2004]
(Ans: 2p – m – 1 )
L6. Given log5 2 = m and log5 7 = p, express
log5 2.82
in terms of m and p.
[4]
(Ans: 2(p + m – 1 )
L7 Given log 2 T - log4 V = 3, express T in terms of
V. [4]
[SPM 2003]
(Ans: T = 8V ½
)
L8. Given log 4 T + log 2 V = 2, express
T in terms of V. [4]
(Ans: 16V-2
)
L9 Solve 42x – 1
= 7x
. [4]
( Ans: x = 1.677 )
L10. Solve 42x – 1
= 9x
. [4]
( Ans: x = 2.409 )](https://image.slidesharecdn.com/5indiceslogarithms-120909011915-phpapp02-150602231856-lva1-app6892/75/5indiceslogarithms-120909011915-phpapp02-12-2048.jpg)
![5 Indices & Logarithms 13
INDICES & LOGARITMS
Enrichment Exercises (Past Year SPM Questions) : PAPER 1
EXERCISE EXCERCISE
1
Solve the equation 2 3
2
1
8
4
x
x
−
+
= .
[SPM 2006 P1,Q6] [3]
(Ans : x = 1 )
2. Given log2 xy = 2 + 3log2 x – log2 y,
express y in terms of x.
[SPM 2006 P1,Q 7] [4]
(Ans: y = 4x )
3 Solve the equation 3 32 log ( 1) logx x+ − = .
[SPM 2006 P1,Q 8] [3]
(Ans : x = 9 / 8 )
4. Solve the equation 4 3
2 2 1x x+ +
− = .
[SPM 2005 P1,Q7] [3]
(Ans : x = -3 )
5 Solve the equation 3 3log 4 log (2 1) 1x x− − = .
[SPM 2005 P1,Q 8] [3]
(Ans: x = 3/2 )
6. Given that log 2m p= and log 3m r= ,
express
27
log
4
m
m
in terms of p and r.
[SPM 2005 P1,Q 9] [4]
(Ans: 3r – 2p + 1 )
7 Solve the equation 2 3 8 6
8 4x x− +
= .
[SPM 2004 P1,Q7] [3]
(Ans : x = 3 )
8. Given that 5log 2 m= and 5log 7 p= ,
express 5log 4.9 in terms of m and p.
[SPM 2004 P1,Q 8] [4]
(Ans: 2p – m - 1 )](https://image.slidesharecdn.com/5indiceslogarithms-120909011915-phpapp02-150602231856-lva1-app6892/75/5indiceslogarithms-120909011915-phpapp02-13-2048.jpg)
![5 Indices & Logarithms 14
INDICES & LOGARITMS
Enrichment Exercises (Past Year SPM Questions) : PAPER 1 (Cont...)
EXERCISE EXCERCISE
1 Solve the equation 2 3 4
16 8x x−
= .
[SPM 2008 P1,Q7] [3]
(Ans : x = -3 )
2. Given log4 x = log2 3, find the value of x.
[SPM 2008 P1,Q 8] [4]
(Ans: x = 9 )
3 Given that log 2 T – log 4 V = 3, express T in
terms of V
[SPM 2006 P1,Q 8] [3]
(Ans : )
4. Solve the equation 2 1
4 7x x−
= .
[SPM 2003 P1,Q6] [3]
(Ans : )
5 Solve the equation 3 3log 9 log (2 1) 1x x− + = .
[3]
(Ans: x = 1 )
6. Given that log 2m p= and log 3m r= ,
express
2
27
log
16
m
m
in terms of p and r.
[4]
(Ans: 3r – 4p +2 )
7 Solve the equation 5 3 6
8 32x x− +
= .
[3]
(Ans : x = 3.9 )
8. Given that 5log 2 m= and 5log 3 p= ,
express 5log 2.7 in terms of m and p.
[4]
(Ans: 3p – m – 1 )](https://image.slidesharecdn.com/5indiceslogarithms-120909011915-phpapp02-150602231856-lva1-app6892/75/5indiceslogarithms-120909011915-phpapp02-14-2048.jpg)
Uploaded bySofia Mahmood
5indiceslogarithms 120909011915-phpapp02
Logarithms and indices are important mathematical concepts. Laws and formulas allow logarithmic and index expressions to be simplified and equations to be solved. Key steps in working with logarithms and indices include using the appropriate laws and formulas to simplify expressions, setting up logarithmic or index equations equal to each other, and solving the resulting linear equations. Practice questions cover a range of skills like evaluating logarithmic expressions without a calculator, expressing logarithmic values in terms of given variables, and solving equations involving logarithms and indices. Mastering the laws, formulas, and problem-solving process is essential for working with logarithms and indices.
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5indiceslogarithms 120909011915-phpapp02
- 1. 5 Indices &Logarithms 1 5. INDICES AND LOGARITHMS IMPORTANT NOTES : UNIT 5.1 Law of Indices I. am x an = am + n II. am ÷ an = am – n III (am )n = amn Other Results : a0 = 1 , m m a a 1 =− , (ab)m = am bm 1. 1000 = 3– 2 = (3p)2 = 5.1.1 “BACK TO BASIC” BIL am × an = am + n am ÷ an = am – n (am )n = amn 1. a3 × a2 = a3 + 2 = a5 a4 ÷ a = a5 – 1 = a 4 (a3 )2 = a3x2 = a6 2. 23 × 24 = 23 + 4 = 2 □ a3 ÷ a5 = a3 – 5 = a □ = 2 1 a (32 )4 = 32x4 = 3 □ 3. p3 × p – 4 = p3 + ( – 4) = p □ = p – 4 ÷ p5 = p – 4 – 5 = p □ = ( ) p 1 (p – 5 )2 = p– 5 x2 = p □ = 4. 2k3 × (2k)3 = 2k3 × 23 × k3 = ( ) k □ (4a)2 ÷ 2a5 = (42 a2 ) ÷ (2a 5 ) = 5 2 2 16 a a = (3x2 )3 = 33 × x2x3 =
- 2. 5 Indices &Logarithms 2 UNIT 5.2 SIMPLE EQUATIONS INVOLVING INDICES No Example Exercise 1 Exercise 2 1. 3x = 81 3x = 34 x = 4 2x = 32 x = 4x = 64 x = 2. 8x = 16 (23 )x = 24 23x = 24 3x = 4 x = 3 4 4x = 32 x = 27x = 9 x = 3. 8x = 16x – 3 (23 )x = 24(x – 3) 23x = 24x – 12 3x = 4x – 12 x = 12 4x+2 = 32x - 1 x = 271 – x = 92x x = 4. 2 × 82x = 16x + 3 21 × (23 )x = 24 (x + 3) 21+3x = 24x + 12 1 + 3x = 4x + 12 1 – 12 = 4x – 3x x = – 11 16 × 42x – 3 = 322 – x x = 251 – 3x = 5 × 125x x = Suggested Steps: S1 : Use the laws of indices to simplify expression (if necessary) S2 : Make sure the base is the SAME S3 : Form a linear equation by equating the indices S4 : Solve the linear equation
- 3. 5 Indices &Logarithms 3 UNIT 5.2 LOGARITMS Do YOU know that .... If a number N can be expressed in the form N = ax , then the logarithm of N to the base a is x? N = ax ⇔ loga N = x 100 = 102 ⇔ log10 100 = 2 64 = 43 ⇔ log4 64 = 3 0.001 = 10-3 ⇔ log10 0.001 = – 3
- 4. 5 Indices &Logarithms 4 Unit 5.2.1To convert numbers in index form to logaritmic form and vice-versa. No. Index Form Logaritmic Form 1. 102 = 100 log10 100 = 2 2. 23 = 8 log2 8 = 3 3. pq = r logp r = q 4. 104 = 10000 5. a3 = b 6. 81 = 34 7. logp m = k 8. 2x = y 9. V = 10x 10. log3 x = y 11. loga y = 2 12. 25 = 32 13. log3 (xy) = 2 14. 10x = y3 15. log10 100y = p
- 5. 5 Indices &Logarithms 5 UNIT 5.2.2 To find the value of a given Logaritm IMPORTANT : No. Logaritmic Form Notes 1. log10 1000 = 3 103 = 1000 dan log10 103 = 3 2. log2 32 = 5 25 = 32 dan log2 25 = 5 3. log10 0.01 = 10 = 0.01 dan log10 = 4. log 4 64 = 4 = 64 dan log4 = 5. log p p = (REINFORCEMENT) 6. log p p8 = log a a2 = 7. log m m-1 = logm 2 1 m = 8. log a a⅓ = log p p-5 = 9. loga 4 1 a = log b bk = 10. log p (p×p2 ) = log p p p = loga ax = x
- 6. 5 Indices &Logarithms 6 UNIT 5.3 Laws of Logaritm I. loga (xy) = loga x + loga y II. loga y x = loga x – loga y III loga xm = m loga x Other Results : loga 1 = 0 (since 1 = a0 ) loga a = 1 (since a = a1 ) NO. Examples Exercises 1. loga 3pr = loga 3 + loga p + loga r (a)loga 2mn = (b) loga 3aq = (c) log10 10yz = (d) log10 1000xy = (e) log2 4mn = 2. loga q p = loga p – loga q (a) loga r p 2 = loga p – loga 2r = loga p – (loga 2 + loga r) = (b) log2 m 4 = (c) log10 kx 10 = (d) log10 100 xy = (e) loga m a3 =
- 7. 5 Indices &Logarithms 7 UNIT 5.3.2 Aplication of the law : loga xn = n loga x 3. Example : loga x 3 = 3 loga x (a) loga 2 1 x = loga x –2 = (b) log2 ( 4 xy ) = log2 x + log2 y 4 = log2 x + (c) log2 ( 4 4y ) = = (d) log2 x y4 = (e) log2 8 4 y = Reinforcement exercises (Laws of Logaritm) 1. Example : log10 100x 3 = log10 100 + log10x3 = log10 102 + 3 log10 x = 2 + 3 log10 x (a) log10 10000x 5 = = = (b) log2 ( 8 4 xy ) = = (c) logp ( 5 8p ) = = (d) log2 3 2 4x k = (e) log4 64 3 y =
- 8. 5 Indices &Logarithms 8 UNIT 5.4 EQUATIONS IN INDICES (Which involves the use of LOGARITM) I. Equation in the form ax = b No. Example Exercise 1 Exercise 2 1. 3x = 18 log10 3x = log10 18 x log10 3 = log10 18 3log 18log 10 10 =x x = 2.631 2x = 9 x = 7x = 20 x = 2. 5x+2 = 16 log10 5x+2 = log10 16 (x+2) log10 3 = log10 18 5log 16log 2 10 10 =+x x+2 = x = 4x+1 = 28 x = 3x-2 = 8 x = 3. 2x+3 = 200 log10 2x+3 = log10 200 x+3 = x = 71-x = 2.8 x = 63x-2 = 66 x = Steps to be followed: S1 : Take logaritm (to base 10) on both sides. S2 : Use the law log10 ax = x log10 a. S3 : Solve the linear equation with the help of a calculator.
- 9. 5 Indices &Logarithms 9 UNIT 5.5 Change of Base of Logarithms Formula : loga x = a x b b log log No. Example Exercise 1 Exercise 2 1. log4 8 = 4log 8log 2 2 = 2 3 (a) log4 32 = 4log 32log 2 2 = (b) log16 8 = (c) log8 2 = = (d) log9 27 = = (e) log81 9 = = (With a calculator) – Change to base 10 1. log4 9 = 4log 9log 10 10 = 1.585 (a) log5 20 = 5log 20log 10 10 = (b) log4 0.8 = (c) log7 2 = (d) log9 77 = (e) log3 9.6 = (f) log6 2.5 = Ans : 0.5114 (g) log5 2000 = Ans : 4.723 (h) log12 6 = Ans : 0.7211
- 10. 5 Indices &Logarithms 10 UNIT 5.6 Aplication of the Laws of Logaritm To Solve Simple Equations involvong logaritms EXAMPLE EVERCISE C1. Solve the equation log2 (x+1) = 3. Answers: log2 (x+1) = 3 x + 1 = 23 x + 1 = 8 x = 7 L1. Solve the equation log2 (x – 3 ) = 2. Jawapan: Ans : x = 7 C2. Solve the equation log10 (3x – 2) = – 1 . Jawapan: 3x – 2 = 10-1 3x – 2 = 0.1 3x = 2.1 x = 0.7 L2. Solve the equation log5 (4x – 1 ) = – 1 . Ans : x = 0.3 L3. Solve the equation log3 (x – 6) = 2. Ans : x = 15 L4. Solve the equation log10 (1+ 3x) = 2 Ans : x = 33 L5. Solve the equation log3 (2x – 1) + log2 4 = 5 . Ans : x = 14 L6. Solve the equation log4 (x – 2) + 3log2 8 = 10. Ans : x = 6 L7. Solve the equation log2 (x + 5) = log2 (x – 2) + 3. Ans : x = 3 L8. Solve the equation log5 (4x – 7) = log5 (x – 2) + 1. Ans : x = 3 L9. Solve log3 3(2x + 3) = 4 Ans : x = 12 L10 . Solve log2 8(7 – 3x) = 5 Ans : x = 1
- 11. 5 Indices &Logarithms 11 UNIT 5.6.1 To Determine the value of a logarithm without using calculator. EXAMPLE EXERCISE C1. Given log2 3 = 1.585, log2 5 = 2.322. Without using a calculator find the value of (a) log2 15 = log2 (3 × 5) = log2 3 + log2 5 = 1.585 + 2.322 = L1. Given log3 5 = 1.465 , log3 7 = 1.771 . Withouf using calculator, evaluate (a) log3 35 = = = = (b) log2 25 = log2 (5 × 5) = = = (b) log3 49 = = = = (c) log2 0.6 = log2 ( 5 3 ) = log2 3 – log2 5 = = (c) log3 1.4 = = = = (d) log2 10 = log2 (2 × 5) = log2 2 + log2 5 = = (d) log3 21 = = = = (e) log4 5 = 4log 5log 2 2 = 2 322.2 = (e) log9 21 = = = = (f) log5 2 = 5log 2log 2 2 = )( 1 = (f) log5 3 = = )( 1 =
- 12. 5 Indices &Logarithms 12 Enrichment Exercises (SPM Format Questions) EXERCISE EXCERCISE L1 Given log3 x = m and log2 x = n. Find logx 24 in terms of m and n. [SPM 2001] [4] (Ans : 3/n + 1/m ) L2. Given log3 x = p and log2 x = q. Find logx 36 in terms of m and n. [4] (Ans: 2/p + 2/q ) L3. Given log3 x = p and log9 y = q. Find log3 xy2 in terms of p and q. [SPM 1998] [4] (Ans: p + 4q ) L4. Given log3 x = p and log9 y = q. Find log3 x2 y3 in terms of p and q. [4] (Ans: 2p + 6q ) L5 Given log5 2 = m and log5 7 = p, express log5 4.9 in terms of m and p. [4] [SPM 2004] (Ans: 2p – m – 1 ) L6. Given log5 2 = m and log5 7 = p, express log5 2.82 in terms of m and p. [4] (Ans: 2(p + m – 1 ) L7 Given log 2 T - log4 V = 3, express T in terms of V. [4] [SPM 2003] (Ans: T = 8V ½ ) L8. Given log 4 T + log 2 V = 2, express T in terms of V. [4] (Ans: 16V-2 ) L9 Solve 42x – 1 = 7x . [4] ( Ans: x = 1.677 ) L10. Solve 42x – 1 = 9x . [4] ( Ans: x = 2.409 )
- 13. 5 Indices &Logarithms 13 INDICES & LOGARITMS Enrichment Exercises (Past Year SPM Questions) : PAPER 1 EXERCISE EXCERCISE 1 Solve the equation 2 3 2 1 8 4 x x − + = . [SPM 2006 P1,Q6] [3] (Ans : x = 1 ) 2. Given log2 xy = 2 + 3log2 x – log2 y, express y in terms of x. [SPM 2006 P1,Q 7] [4] (Ans: y = 4x ) 3 Solve the equation 3 32 log ( 1) logx x+ − = . [SPM 2006 P1,Q 8] [3] (Ans : x = 9 / 8 ) 4. Solve the equation 4 3 2 2 1x x+ + − = . [SPM 2005 P1,Q7] [3] (Ans : x = -3 ) 5 Solve the equation 3 3log 4 log (2 1) 1x x− − = . [SPM 2005 P1,Q 8] [3] (Ans: x = 3/2 ) 6. Given that log 2m p= and log 3m r= , express 27 log 4 m m in terms of p and r. [SPM 2005 P1,Q 9] [4] (Ans: 3r – 2p + 1 ) 7 Solve the equation 2 3 8 6 8 4x x− + = . [SPM 2004 P1,Q7] [3] (Ans : x = 3 ) 8. Given that 5log 2 m= and 5log 7 p= , express 5log 4.9 in terms of m and p. [SPM 2004 P1,Q 8] [4] (Ans: 2p – m - 1 )
- 14. 5 Indices &Logarithms 14 INDICES & LOGARITMS Enrichment Exercises (Past Year SPM Questions) : PAPER 1 (Cont...) EXERCISE EXCERCISE 1 Solve the equation 2 3 4 16 8x x− = . [SPM 2008 P1,Q7] [3] (Ans : x = -3 ) 2. Given log4 x = log2 3, find the value of x. [SPM 2008 P1,Q 8] [4] (Ans: x = 9 ) 3 Given that log 2 T – log 4 V = 3, express T in terms of V [SPM 2006 P1,Q 8] [3] (Ans : ) 4. Solve the equation 2 1 4 7x x− = . [SPM 2003 P1,Q6] [3] (Ans : ) 5 Solve the equation 3 3log 9 log (2 1) 1x x− + = . [3] (Ans: x = 1 ) 6. Given that log 2m p= and log 3m r= , express 2 27 log 16 m m in terms of p and r. [4] (Ans: 3r – 4p +2 ) 7 Solve the equation 5 3 6 8 32x x− + = . [3] (Ans : x = 3.9 ) 8. Given that 5log 2 m= and 5log 3 p= , express 5log 2.7 in terms of m and p. [4] (Ans: 3p – m – 1 )