Post on 03-Jun-2018
8/12/2019 Indices Log
1/18
ndices&logarithmsindices&logarith
msindices&logarithmsindices&logarit
hmsindices&logarithmsindices&logar
thmsindices&logarithmsindices&logarithmsindices&logarithmsindices&l
ogarithmsindices&logarithmsindices
&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsind
ces&logarithmsindices&logarithmsin
dices&logarithmsindices&logarithmsi
ndices&logarithmsindices&logarithmsindices&logarithmsindices&logarith
msindices&logarithmsindices&logarit
hmsindices&logarithmsindices&logar
thmsindices&logarithmsindices&log
arithmsindices&logarithmsindices&l
ogarithmsindices&logarithmsindices
&logarithmsindices&logarithmsindices&logarithmsindices&logarithmsind
INDICES&
LOGARITHMS
Name
........................................................................................
8/12/2019 Indices Log
2/18
Indices and Logarithms
zefry@sas.edu.my 2
CHAPTER 5 : INDICES AND LOGARITHMS
1.1 Finding the value of number given in the form of :-
Type of indices In General Examples(a)Integer indices (i) positive indices
aaaaan .....
nfactors
a = base(non zero number)
n = index(positive integer)
53 3 3 3 3 3
2( 4)
3)2.0(
31
5
(ii) negative indices
n
n
aa
1
12
=1
2
23
2
4
(b)Fractional indices(i) nn aa
1
n = positive integer
a 0
21
4 4 2
41
16 [2]
1
532 [2]
(ii) mnn mn
m
aaa 32
27 2 23( 27) 3 9
2
38 [4]
3
416 [8]
Notes: Zero Index : 0,10 awherea
Examples :
00 0 0 1
5 1, 2.2 1, ( 3) 1, 12
8/12/2019 Indices Log
3/18
Indices and Logarithms
zefry@sas.edu.my 3
ACTIVITY 1:
Find the value for each of the following;
(a)1
264 64 8 (b) 31
8
[2]
(c) 41
16
[2]
(d) 52
32
[4]
(e)2
327
[9]
(f)1
225
[1
2]
(g)
1
31
8
[2]
(h)
2
14
[16]
(i)
11
32
[32]
(j)
21
3
[9]
EXERCISE 11. Evaluate the following:
(a)3
24
[8]
(b) (16)1
4
[2]
(c)
11
25
[25]
(d)
1
532
81
[2
3]
(e) 1 2(5 )
[ 125
]
(f)1
24
[1
2]
2. Write in index form
(a)1
p
[p-1
]
(b)3
1
q
[q-3
]
(c)
1
1
p
[p1]
8/12/2019 Indices Log
4/18
Indices and Logarithms
zefry@sas.edu.my 4
ACTIVITY 2:
1. Simplify each of the following:
(a) 52 aa (b)2 35 5n n
(c)6
2
x
x
(d)34 2 2n n n
(e) 1
6 4 2p q (f) 1
4 12 4a b
(g) 1
8 2 281p q
(h) 416 2n
(i) 2 33 6 2
(j)
1 1
5 3
32 125
(k) 1 12 4 2n n (l)
2 3
2 4
n na a
a a
LAWS OF INDICES
nmnm aaa nmnm aaa mnnm aa )(
mmm baab )( n
nn
b
a
b
a
2 5
7
a
a
5[5 ]
n
[1] 4[ ]x
3 2[ ]p q 3[ ]ab
4 1[9 ]p q 8[2 ]n
5 4[3 2 ] 2
[ ]5
[4] 5 6[ ]na
8/12/2019 Indices Log
5/18
Indices and Logarithms
zefry@sas.edu.my 5
2. Prove that
(a) 11 4344 nnn isdivisible by 17
for all positive integers of n.
17(4n-1) is a multiple of 17 and hence,
17(4n-1) is divisible by 17.
(b) 21 555 nnn is divisible by 31 for
all positive integers of n.
[31(5n) is a multiple of 31 and hence, 31(5
n) is
divisible by 31].
(c) 21 333 nnn is divisible by 13 for
all positive integers of n.
[13(3n) is a multiple of 13 and hence, 13(3
n) is
divisible by 13].
(d) 2 32 2p p is divisible by 12 for all
positive integers ofp.
[12(2p) is a multiple of 12 and hence, 12(2
p) is
divisible by 12].
1
1
1
44 4 4 3
4
34 (4 1 )
4
174
4
17(4 )
nn n
n
n
n
8/12/2019 Indices Log
6/18
Indices and Logarithms
zefry@sas.edu.my 6
2. LOGARITHMS AND THE LAW OF LOGARITHMS.
_____________________________________________________________________
2.1 Express equation in index form to logarithm form and vice versa
Definition of logarithm
If a is a positive number and a 1 , then
`
(INDEX FORM) (LOGARITHM FORM)
N= Number
a= base
x= index
We can use this relation to convert from index form to logarithm form or vice versa.
ACTIVITY 3:1. Convert each of the following from index form to logarithm form:
INDEX FORM LOGARITHM FORM
(a) 4 = 644
log 64 3
(b) 3 =81
(c) 2-3 = 18
(d)10-2 = 0.01
(e)1
3= 13
2. Convert each of the following from logarithm form to index form:
LOGARITHM FORM INDEX FORM
(a) log7 49 = 2 2
49 7
(b) log3 27 = 3
(c) log9 3 =1
2
(d) log10 100 = 2
(e) log51
16= - 4
Notes! Since a1 = a then loga a = 1 Since a0= 1 then loga 1 = 0
xaN xNa log
LogaNis read as logarithm ofN
to the base a
8/12/2019 Indices Log
7/18
Indices and Logarithms
zefry@sas.edu.my 7
3. Find the value of x .
a) log2x= 112 2x
b) log10x= -3
[0.001]
c) log3x= 4
[81]
d) logax= 0
[1]
2.2 Finding logarithm of a number
Logarithm to the base of 10 is known as the common logarithm. The value of
common logarithms can be easily obtained from a scientific calculator .
In common logarithm, if log10N =x, then , antilogN= 10x. [lg N = log10N]
ACTIVITY 41. Use a calculator to evaluate each of the following;
(a) log10 16 = 1.2041 (b) log10 0.025 =
[-1.6021]
(c) log102
3
=
[-0.1761]
(d) log1052 =
[1.3979]
(e) antilog 0.1383 =
[1.3750]
(f) antilog (- 0.729) =
[0.1866]
(g) antilog 1.1383 =
[13.7450]
(h) 10- =
[0.01]
2. Find the value of the following logarithms.
(a) log416= 2
4
log 4 2 (b) log327
[3]
(c) log21
2
[-1]
(d) log82
[3]
(e) log22
[3]
(f) logaa
[4]
8/12/2019 Indices Log
8/18
Indices and Logarithms
zefry@sas.edu.my 8
2.3 Finding logarithm of a numbers by using the Laws of Logarithms
ACTIVITY 5:
1. Evaluate each of the following without using calculator.
(a) log 232= 5
2log 2 5 (b) log327
[3]
(c) log 31
[0]
(d) log 39
[2]
(e) log 864
[2]
(f) log28
[3]
2. Find the value of
(a) log 26 + log 212log 218
2
2
6(12)log
18
log 4
2
(b) log 318 + 2log 36log 372
[2]
LAWS OF
LOGARITHMS
logaxy = logax + logay
logaxm= m logax
y
xa
log = logax - logay
8/12/2019 Indices Log
9/18
Indices and Logarithms
zefry@sas.edu.my 9
(c) 2log42 - 4log 3+ log4 12
[2]
(d) log 545+ 5 5 5log 100 log 10 log 18
[2]
3.. Given that log 23 = 1.585 and log 25 = 2.322 . Evaluate each of the following.
(a) log 215
[3.907]
(b) log 275
[6.229]
(c) log 220
[4.322]
(d) log21.5
[0.585]
5. Given that log32 =0.6309 and log35 = 1.4650. Evaluate each of the following.
(a) log 310
[2.0959]
(b) log318
[2.6309]
(c) log 345
[3.4650]
(d) log30.3
[-1.0959]
8/12/2019 Indices Log
10/18
Indices and Logarithms
zefry@sas.edu.my 10
2.4 Simplifying logarithmic expressions to the simplest form .
ACTIVITY 6
1. Express each of the following in terms of log a, log band/or log c.
(a) log ab
[log a+ log b]
(b) log 3 2
a b
[3log a+ 2log b]
(c) log 3
2ab
[3log a+6log b]
(d) logab
c
[log a+ log b-logc]
2. Express each of the following in term of log axand logay.
(a) log axy
[log log ]a a
x y
(b) log ax y
[2log 3log ]a a
x y
(c) loga
2
xy
[2log log ]a a
x y
(d) log a
2 3
a xy
[2 3log log ]a a
x y
8/12/2019 Indices Log
11/18
Indices and Logarithms
zefry@sas.edu.my 11
3. Write each of the following expressions as single logarithm:
(a) lg 3 + lg 25 = lg 3 + lg 5
= lg 15
(b) 3 lg 2 + 2 lg 32 lg 6
[log 2]
(c) log2x+ log2y
[ 2
2log xy ]
(d) lg 6 + 2 lg 4lg 8
[log 12]
(e) lgx+2lgy- 1
[
2
log10
xy
]
(f) 3 lgx2
1lgy4+ 2
[
3
2
100log
x
y
]
(g) 2logax - 1 + loga y
[2
logax y
a
]
(h) log3x+ 2log3y1
[
2
3log3
xy
]
(i) logbx+ logby+ 1
[ logb xyb ]
(j)log ax + logay1
[ logaxy
a
]
8/12/2019 Indices Log
12/18
Indices and Logarithms
zefry@sas.edu.my 12
3. CHANGE OF BASE OF LOGARITHMS_________ __________________________________________________________________
3.1 Finding logarithm of a number by changing the base of the logarithm to asuitable base
The base of logarithms can be changed to other base by using a formula :
ACTIVITY 7:1. Find the value of each of the following. Give your answer correct to four
significant figures.
(a)
5log 2
(b) 8log3
[1.893]
(c)3
log 4
[1.262]
(d) 5.0log 2
[-1.00]
2. Find the value of each of the following without using calculator..
(a) log 2 9. 3log 8 (b) log 3 7. log 7 2. log 2 3 =
[1]
(c)4
log 16 . log 3 125 =
[8]
(d) log 4 5. log 5 3. log 37. log 7 64 =
[3]
a
bb
c
c
alog
loglog
When c= b, thena
bb
b
b
alog
loglog
=a
blog
1
log2
log5
0.3010
0.6990
0.431
3 3
3 3
log 9 log 8
log 2 log 3
2 3 6
8/12/2019 Indices Log
13/18
Indices and Logarithms
zefry@sas.edu.my 13
3.2 Solving Problems involving the change of base and laws of logarithms.
ACTIVITY 81. Given that log 23 = 1.585 and log 2 5 = 2.322.Find the value of each of the
following.
(a) log3 15 (b) log 33
5
[2.4650] [-0.4650]
2. Given that log2 a = b. Without using the calculator, express the following in termsof b:
(a) loga 16 (b) log 16a
[4
b] [
4
b]
(c) log 4 a (d) log a32a
[2
b] [
51
b ]
3. If log 3x= rand log 3y=s, express each of the following in terms of rands.
(a) log 3x2y (b)
3
9log
x
y
[2r+s] [2+r-s]
4. If mx2log and ny2log , express each of the following in terms of mand n.
(a)4
log xy (b) 2logx
y
[ 1[ ]2
m n ] [ 2nm]
8/12/2019 Indices Log
14/18
Indices and Logarithms
zefry@sas.edu.my 14
4. EQUATION INVOLVING INDICES AND LOGARITHMS
4.1 Solving equations involving indices
METHOD:1. Comparison of indices or base
(i) If the base are the same , when yx aa , then x = y(ii) If the index are the same , when xx ba , then a = b
2. Using common logarithm
ACTIVITY 9:1. Solve the following equations:
(a) 3x= 81 (b) 16x= 8
[3
4]
(c) 8x+ = 4
[1
3
]
(d) 9x+ = 3
[1
2
]
(e) 9x. 3x- = 81
[5
3]
(f) 2x+ - 4 x = 0
[1]
(g) 814 x
[ 3 ]
(h) 3
125
1 x
[5 ]
43 3
4
x
x
8/12/2019 Indices Log
15/18
Indices and Logarithms
zefry@sas.edu.my 15
2. By using common logarithm(log10), solve the following equations and give your
answer correct to two decimal places.
(a) 2x= 3 (b) 87 x
[-1.07]
(c) 4 x+ = 7
[0.2]
(d) xx 3.2= 18
[1.61]
(e) 46 25 x
[2.11]
(f) 493.2 xxx
[21.68]
3. By using replacement method, solve each of the following equations:
(a) 122 34 xx (b) 433 21 xx
[x= -1]
(c) 13 3 9x x
[x= 2]
(d) 3 22 2 12x x
[x= 0]
lg2 lg3lg3
1.58lg2
x
x
4 32 2 2 2 1
2
16 8 1
8 1
1
8
x x
xlet y
y y
y
y
3
2
12
8
2 2
3
x
x
x
Substitute y
x
8/12/2019 Indices Log
16/18
Indices and Logarithms
zefry@sas.edu.my 16
4.2 Solving Equations involving logarithms
METHOD:
1. For two logarithms of the same base, if nm aa loglog , then m = n.
2. Convert to index form, if nma log , then m = an.
ACTIVITY 10:1. Solve the following equations.
(a) lgx = lg 3 + 2 lg 2lg 2 (b) 2 lg 3 + lg (2x) = lg (3x+ 1)
[1
15]
(c) lg (4x3) = lg (x+ 1 ) + lg 3
[ 6 ]
(d) lg (10x+ 5)lg (x+ 4 ) = lg 2
[3
8]
2. Solve the following equations :
(a) lg 25 + lgxlg (x1) = 2 (b) lg 4 + 2 lgx= 2
[5]
23 2
lg2
6x
2
25lg 2
1
25 10 ( 1)
25 100 100
75 100
4
3
x
x
x x
x x
x
x
8/12/2019 Indices Log
17/18
8/12/2019 Indices Log
18/18
Indices and Logarithms
f @ d 18
SPM QUESTIONS
1. SPM 2003
Given that 3loglog 42 VT , express T in terms of V .[4 marks] [T= 8 V ]
2. SPM 2003
Solve the equation xx 74 12 . [4 marks] [x= 1.677]
3. SPM 2004
Solve the equation 684 432 xx . [3 marks] [x= 3]
4. SPM 2004
Given that m2log5 and p7log5 , express 9.4log5 in terms of mandp.
[4 marks] [2pm-1]
5. SPM 2005
Solve the equation 1)12(log4log 33 xx . [3 marks] [2
3x ]
6. SPM 2005
Solve the equation 122 34 xx . [3 marks] [x= - 3]
7. SPM 2005
Given that pm 2log and rm 3log , express
4
27log
mm
in terms ofpand r.
[4 marks] [3r2p+ 1]
8. SPM 2006
Solve the equation 2 32
18
4
x
x
. [3 marks] [x =1]
9. SPM 2006
Given that 2 2 2log 2 3log logxy x y , expressyin terms ofx.
[3 marks] [y=4x]
10. SPM 2006
Solve the equation3 3
2 log ( 1) logx x . [3 marks] [ 1
18
x ]