SPM Add Math Form 4 Chapter 5 Indices & Logarithms

SPM Add Math Form 4 Chapter 5 Indices & Logarithms CHAPTER 5 : INDICES AND LOGARITHMS

Indices and Laws of Indices Integer and fractional indices 1) Find the value of

2 3

3

3 (c)   4

0

(c)  

(b) (2) 5

(a) 3²

(d) (0.5) 4

2) Find the value of 0

(b) (7)

(a) 6

0

(d) (2.5) 0

3) Evaluate the following. (a) 2

4

(b) (4)

2

 1 (c) 1   4

3

(d) (0.1) 1

4) Evaluate the following.

(a) 9

1 2

(b) (64)

1 3

1

(c)

0.0625

1 4

 8 3 (d)    27 

5) Evaluate the following.

(a) 32

4 5

(b) (8)

2 3

3

 27  (d)    125 

 16  4 (c)    81 



2 3

Law of indices 6) Simplify the following. 2

(a) 2 3 x 2 4

7

(b) (3) 2 x (-3) 6

(c) 9 3 x 9 3

(b) 6 4  6 -3

(c) 4 4  4

(d) 7 3n x 7 n x 7 3

7) Simplify the following. 3

(a) 85  83



9 4

(d)

5 2 n x 5 3n 5 4n

(d)

243 5

8) Simplify the following. 2 3

(a) (2 )

3 4 8

(b) (5 )

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 

(c) a 3

2

4

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms 9) Simplify and evaluate the following.

(a) 53 x 25 2  125

(b)

4x8 32

1 3

2

(c)

4 5

3 x9 27

1 3

(d)

2 3

8



2 3

x4 23

1 2

Simplifying algebraic expressions 10) Simplify each of the following. (b) 36 n  32n

(a) 2 3n x 2 n

(c) (4 2 n ) 3

a x a3

(d)

11) Simplify each of the following. (b) (32 p 5 q -10 )

(a) (m 3 n 2 ) 4



2 5

(c)

12a 5 b 2 4a 3 b 6

(d)

3

m 3 n 6

12) Simplify each of the following. 1

1

n

1

(a) 4 n x 43-n  43n1

(b) ( p 2 n ) 2  ( p 6 ) 3 x ( p 4 n ) 4

a 3n  2 (c) n 1 a x a 2 n4

(2m 2 ) 3 x 3m 3 (d) 6m 2

13) Simplify each of the following. (a) 2 n 2 x 4 n  83n

(b) 9 2 x x 27 x -3  34-x

(c) 5 x6 x 252x-1  1252-x

(d) 16 2n  23n x 8 2

14) Show that 4 3n 

64 2 2n

15) If 2x = y, express the following in terms of y. (a) 2 3 x

(b) 4 x 1

(c) 8(4 x 2 )

(d) 8 x - 4 -x

16) Show that 2n + 2n+1 + 2n+2 is divisible by 7 for all positive integers of n. 17) Given that 2a = 4b = 8c , express c in terms of a and b. 18) Simplify 3n+2 - 3n - 27(3n-1) in the form k(3n ) where k is a constant. Hence, write down the value of k.

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms Logarithms and Laws of Logarithms Expressing equation in index form to logarithm form and vice versa 19) Convert the following to logarithm form. (a) 2 2  4

(b) 53  125

(c) 3  2 

1 9

(d) 3 p  q

20) Convert the following to index form. (a) log 2 32  5

(b) log 3 9  2

(c) log 3 3 

1 2

1 4

(d) log 2    2

21) Find the value of x in each of the following equations. (a) log 3 x  1

(b) log 2 8  x

(d) log 3 x  2  2

(c) log x 9  1

Finding logarithm of a number 22) Find the value of each of the following. (a) log 2 64

(b) log 1 9

(c) log 8 0.25

(d) log 5  7

(e) log 4 2

(f) log 1 1

3

23) Find the following logarithms using a scientific calculator. (a) log 10 0.4

(b) log 10 5.25

(c) log 10 35

1 2

(d) log 10  

Laws of logarithms 24) Given that loga 2 = 0.301 and loga 3 = 0.477, find the value of (a) log a 6

(b) log a 1.5

(c) log a 8

25) Given that log10 x = p and log10 y = q, express the following in terms of p and q. (a) log 10 ( x 2 y)

 10 y    x 

(b) log 10 

(c) log 10 10 xy 3

 100 x    y2   

(d) log 10 

26) Given that x = 5a and y = 5b , express the following in terms of x and y. (a) log 5 xy 2

 x   5y   

(b) log 5 

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 1    xy   

(c) log 5 

(d) log 5

x2 y3

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms 27) Evaluate the following. (a) log 4 4 4

(b) log 2

1 4

(c) log 2 8

(d) log 27 3

(e)

log a 27 log a 9

(f) log 9 3

28) Evaluate the following logarithms without using a calculator. (a) log 8 4  log 8 2

(b) log 3 36 - log 312

(c) log 7 4  2 log 7 3  2 log 7 6

(d) 2 log 2

2 81 3  log 2 - 2 log 2 3 8 4

 p   q  in terms of m and n. .   30) Given that log 2 5  m and log 2 6  n , express log 2 3.6 in terms of m and n. . 29) Given that log 2 p  m and log 2 q  n , express log 2  8

Logarithmic expressions in the simplest form 31) Simplify each of the following as a single logarithm. (a) log 10 a  2 log 10 b  3 log10c (c)

(b) 2 log a 6 - 2 log a 3

1 1 log a x  2 log a y  log a z 2 3

(d)

1 log 5 ( x  1) - 2 log 5 ( x  1) 2

Change of Base of Logarithms Changing the base of logarithms 32) Find the values of the following. (a) log 3 10

(b) log 3 15

(c) log 0.5 8

33) Find the value of the following without using a calculator.. (a) log 9 27

(b) log 4

(c) log 64 8

2

34) Evaluate the following. (a) 4 log 3 5 x 2 log 5 3

(b) log 7 5 . log 5 9 . log 9 7

(c) log 4 5 . log 5 6 . log 6 7 . log 7 8

35) Given that log4 3 = 0.792 and log4 5 = 1.161, evaluate the following. (a) log 4 0.6

(b) log 3 4

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(c) log 5 16

(d) log 5 3

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms 36) Given that log2 x = m and log2 y = n, find the following in terms of m and / or n. (b) log y

(a) log x 2

2

(d) log 4 xy

(c) log y x

37) Given that log2 5 = 2.322 and log2 7 = 2.807, find the value of the following without using a calculator.

 25    7 

(b) log 4 

(a) log 2 175

38) Evaluate

log 9 36 x log 49 9 log 7 6

Problems involving the change of base and laws of logarithms 39) Given that a b = 81, find log9 a in terms of b. 40) If log4 x = k, express each of the following in terms of k. (b) log 2 8 x 2

(a) log 8 x

41) If log10 5 = p, express each of the following in terms of p. (a) log 5 2

(b) log 10 2

42) Given that log2 x = m and log2 y = n, express each of the following in terms of m and n. (a) log x 2  log y 2

(b) (log x 8)(log 8 y)

(c) log xy 4

(d) log 4

xy 3

43) Evaluate the following expressions. (a) log 4 64  log 9 27  log 5 25

(b) log 3 2  log 9 36  log 1 27 3

44) Simplify each of the following. (a) log 2 8m  log 8 m 3

(b) log 9 9 x 4  log

3

x

45) Given that log3 a = p and log5 a = q, express loga 75 in terms of p and q.

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms Equations involving Indices and Logarithms Solving indicial questions 46) Solve the following equations. (a) 16 x 

1 32

(b) 9 2 x 1  27 x

(c) 2 x x 4 x -1  8 2x-1

(d) 53x  25 x 1 

1 25

47) Solve the following equations. (a) x 2  64

(b) 3x 3  24

(c) x -5 

1 32

(d) ( x  2) 4  81

48) Solve the following equations. (b) 3 x 1  0.45

(a) 3 x  7

(c) 2 x . 3x  5 x 1

(d) 6 x 1  5 2 x 1

49) By using substitution y = 3 x , find the value of x such that 9 x +3 = 4(3 x). 50) Given that y= ax b -4 and that y=12 when x=2 and y=50 when x=3, find the values of a and b. 51) Solve the following simultaneous equations.

3 9   27 x

2y

2x 1  4y 8

52) A liquid cools from its original temperature 80°C to a temperature T°C in x minutes. Given that T= 80(0.98)x , find the value of (a) T when x = 30 (b) x when T =50 53) Solve the equation 32x-1 = 5x . 54) Solve the equation 812x = 93x+1 . 55) Solve the equation 2x+3 - 2x+2 =16. Solving logarithmic equations 56) Solve the following equations. (a) log 3 ( x  2)  log 3 (4 x  11)

(b) 3 log 10 ( x  1)  log 10 8

(c) log 2 4 x  2 log 2 5  3

(d) 3 log x 2  log x 4  5

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SPM Add Math Form 4 Chapter 5 Indices & Logarithms 57) Solve the following equations. (a) log 3 2  log 9 ( x  2)

(b) log 5 x  4 log x 5

(c) log 4 x  log 2 x  3

(d) log 3 x  4 log x 3  4

58) Given that log4 3y = 2 log4 x + 1, express y in terms of x . 59) Solve the simultaneous equations.

3x  27 3y

log 10 ( x  2 y )  log 10 15

60)

(a) Solve 2 log3 x  32 (b) If 2 3x  9(3 2 x ), prove that x log a

8  log a 9. 9

Given 2 log 5 xy  2  log 5 ( x  1)  log 5 y when x and y are both positive, show that 61)

y

25(x  1) x2

62) Solve the simultaneous equations.

4x 2 2y

log 10 (2 x  2 y)  1

63) Given that log3 T – log9 V = 2, express T in terms of V . 64) Solve the equation log4 3x – log4 (2x-1) = 1.

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