Index Log Ver.2012
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Transcript of Index Log Ver.2012
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My
AdditionalMathematicsModules
Form 4Topic 5
Haiya manyak sinang punyama...!
(Version 2012)
by
NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)
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3125
5.1 INDICES AND LAWS OF INDICES (2)
IMPORTANT NOTES:
1. For an index numberan, (read as a raise to the power ofn), where a is the base and n is the index.
2. ao
= 1 3. na
1
= a n
, where a 0 4. a n1
=n
a , where a 0, (read as a raise to the nth
root). 5. a nm
=
n ma
6. Laws of Indices
6.1 am x an = am + n 6.2 am an = amn (or)a
a
n
m
=nma 6.3 (am)n = am x n
6.4
m
b
a
=
m
m
b
a6.5 (ab)m = am bm
7. Equation Involving Indices
Solve the problem 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 =alog
blogm
1. Evaluate each of the following without using a calculator.
(a) 43 (b) 24(c)
273
2
(d)
(e)
3
4
3
(f) 0.50 (g) 31
8
(h) 7 0
2. Simplify and then evaluate each of the following
(a) 32 x 35 (b) 32 34 (c) (52)3 (d) 43 x 24 162
(e) 43 x 45 (f)3
2
8
4 (g) 32
27x 9
3
2
(h) (5
2)2 (125)
Exercise 5.1
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3. Simplify each of the following expressions. (3)
(a) 2 m + 3 x 4 m 32m (b) 3 n + 2 3 n 1 (c) 92k 3 k+ 1 x 27 k
(d) 25n x 42n x 63n (e) 20a3 5a-5(f)
)1m(2
1m2m
9
813
+
+
4. Solve each of the following.
(a) Show that 23w + 1 = 2(8w) (b) Show that 5 n + 1 + 5 n 3(5cn 1) is divisible by 3 or 9.
(c) Show that 22x + 3 (9x + 1 32x) = (3
2)2x
(d) Express 3 m 2 + 3 m + 1 2(3 m) in the simplest form.
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(e) Express 9 x + 1 32x + 12(32 x 1) in the simplest form. (f) Show that 7p + 1 + 7p + 2 is a multiple of 8.
5.2 LOGARITHMS AND LAWS OF LOGARITHMS
(4)
IMPORTANT NOTES:
1. To convert an equation in index form to logarithm form and vice versa.
IfN = 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 logaxy = loga x + logay
4.2 logay
x
= logax loga y (or) loga (x y) = logax loga y
4.3 logaxm
= m logax
5. Change of Bases of Logarithms:
5.1 logab =alog
blog
c
c5.2 logab =
alog
blog
b
b=
alogb
1
6. Equations Involving Logarithms:
6.1 Converting the equation of logarithm to index form, i.e.
logaN = 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.
1. Express the following equations to logarithm form or index form.
(a) 32 = 25 (b) 4 =83
2 (c) logxq = p (d) 1 = 100
(e) 3 = log327
1
(f) log3243 = 5 (g) px= 5 (h) log636 = 2 (i) x
3 = 108 (j) 2 = log5 25
Exercise 5.2
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x
64
1log
3 m
2. Determine the value of x in each of the following equations.
(a) log3 81 =x (b) log4x =2
1(c) logx 125 = 3
(d) log2x = 2 (e) =3
1(e) logx
216
1= 3
3. Find the value of each of the following. (5)
(a) log10 100 = (b) log10 39.94 = (c) antilog 1.498 =
(d) log10 35
1
=
(e) antilog 0.3185 = (f) antilog (0.401) =
4. Find the value of each of the following without using a calculator.
(a) log2 32 (b) log3243
1 (c) log9 9
(d) log5 0.2 (e) log749 (f) logm
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 (3
125)
(d) log2 1.5 (e) log2 0.6 (f) log2 20
6. Simplify each of the following expression to the simplest form.
(a) 2 log2x log2 3x + log2y (b) loga 5x + 3 loga 2y (c) logbx + 3 logbx + logb (y + 1)
(d) log2 4x log2 3y 2 (e) 3 log3x + log3 3y + log3 2x (f) log5 m log5 4m log5 k
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7. Determine the values of the following logarithms.
(a) log2 7 (b) log3 23 (d) log0.5 8.21 (c) log3 5
8. Given that log2 w = p, express the following in terms ofp. (6)
(a) logw 4 = (b) log8 16w2 =
(c) log432
w (d) log w 64
9. Given that logm 3 = xand logm 4 = y. Express the following in terms ofxand/ory.
(a) log36m = (b) log3m12
(c) log3 m16 (d) log34
m
Exercise 5.3
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Exercise 5.4 SPM QUESTIONS ( 2003 2010 )
1. Solve the equation 82x 3 =
4
1
2x[3 marks]
SPM2006/Paper1
(8
)2. Given that log2 xy = 2 + 3 log2 x log2y, express y in terms ofx. [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. [3 marks]SPM2005/Paper1
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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 logm ( 427m
) in terms ofp and r. [4 marks]
SPM2005/Paper1
(9)
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
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10. Given that log2 b =x and log2 c = y, express log4
c
8b
in terms ofx andy. [4 marks] SPM2007/Paper
1
11. Given that 9(3n1) = 27n, find the value ofn. [3 marks] SPM2007/Paper 1
(10)
12. Given that log4x = log2 3, find the value ofx. [3 marks] SPM2008/Paper
1
13. Solve the equation: 162x 3 = 84x [3 marks] SPM2008/Paper
1
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9
8
14. Given that 3n 3x 27n = 243, find the value of n. [3 marks] SPM2009/Paper
1
15. Given that log8p log2 q =0, expressp in terms ofq. [3 marks] SPM2009/Paper 1
(11
)
16. Solve the problem: 3x + 2 = 3x = [3 marks] SPM2010/Paper
1
17. Given log2 3 = a and log2 5 = b, express log8 45 in terms ofa and b. [3 marks] SPM2010/Paper1
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18. Solve the problem: 2 3x = 8 + 2 3x 1 [4 marks] SPM2011/Paper
1
19. Given log2x = h and log2y = k, express log2y
x3
in terms of h and k. [3 marks] SPM2010/Paper
1
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