Logarithma SPM

1. 1 log a p = x , log a q =y , log a = z
Find log a
𝑝2
𝑞
=log a
𝑝2
𝑞
=log a p2
– log a qr
=2x –( log a q + log a r )
=2x –( y + z )
=2x-y-z
2 Solve the equation 9 x
= 9 ( 3x
)-18
32x
= 32
(3x
) - 18
32x
= 32
(3x
) - 18
(3x
)2
= 32
(3x
) -18
Gantikan Y = 3x
Y2
= 9y - 18
Y2
= 9y +18=0
(y-3) ( y-6)=0
Y=3
3x
=3 3X
=6
X=1 log 10 3x
= log 10 6
x log 10 3 = log 10 6
X = log 10 6/ log 10 3 =1.631
3 Given a2
=p,
√𝑎=q
Log a
𝑝2𝑞
𝑎
Solution
a2
=p
log a a2
= log a p
2= log a p
√𝑎=q
a= q2
log a a= log a q 2
½ = log a q

2. Log a
𝑝2𝑞
𝑎
= log a p2
q – log a a
=( log a p2
+ log a q) - 1
=2 log a p + log a q -1
=2(2) + ½ - 1
=3 ½
4 log a2=m log a 3= n
log 6 24
= log a 24 / log a 6
= log a ( 23
x 3) / log a ( 2 x 3)
= log a 23
+ log a 3 / (log a 2 + log a 3)
= 3 m + n /( m + n)
5 Solve the equation 6. Log a (
𝑥3 𝑦4
𝑧
)
= (𝑙𝑜𝑔 𝑎 𝑥3
+ 𝑙𝑜𝑔 𝑎 𝑦4
) − 𝑙𝑜𝑔 𝑎 𝑧
= (3 𝑙𝑜𝑔𝑎 𝑥 + 4 𝑙𝑜𝑔 𝑎 𝑦 ) −
𝑙 𝑜𝑔 𝑎 𝑧
Log 10 x = 9 log x 10
Log 10 x = 9 log x 10
= 9 log 10 10
Log 10 x
7 Log 2 64
= log 2 26
= 6 log 22
= 6 (1)
(log 10 x) ( log 10 x ) = 9
(Log 10 x)2
= 9
Log 10 x = √9
= 3
x = 10 3
= 1000
8 Log 9 3
= Log 3 3/log 3 9
=1/ log 3 32
=1/2 (log 3 3)
=1/2 (1) =1/2

3. 9 𝑙𝑜𝑔3 𝑥 = 𝑙𝑜𝑔9121
𝑙𝑜𝑔3 𝑥
=
𝑙𝑜𝑔3 121
𝑙𝑜𝑔39
𝑙𝑜𝑔3 𝑥 ( 𝑙𝑜𝑔39 ) = 𝑙𝑜𝑔3 121
𝑙𝑜𝑔3 𝑥 ( 𝑙𝑜𝑔3 32 = 𝑙𝑜𝑔3 121
(𝑙𝑜𝑔3 𝑥 ) 2( 1 ) = 𝑙𝑜𝑔3 121
𝑙𝑜𝑔3 𝑥 2
= 𝑙𝑜𝑔3 121
𝑥 2
= 121
X = √121
= 11
10 . Express R 11. 0.4 𝑥+3
= 0.52𝑥
𝑙𝑜𝑔3 𝑅 − 𝑙𝑜𝑔9 𝑇 = 2 lg 0.4 𝑥+3
= 𝑙𝑔0.52𝑥
𝑙𝑜𝑔3 𝑅 −
𝑙𝑜𝑔3 𝑇
𝑙𝑜𝑔3 9
= 2 ( 𝑥 + 3) 𝑙𝑔0.4 = 2𝑥𝑙𝑔 0.5
𝑙𝑜𝑔3 𝑅 −
𝑙𝑜𝑔3 𝑇
𝑙𝑜𝑔3 32 = 2 𝑥𝑙𝑔0.4 + 3𝑙𝑔0.4 = 2𝑥𝑙𝑔0.5
𝑙𝑜𝑔3 𝑅 −
𝑙𝑜𝑔3 𝑇
2(1)
= 2 𝑥𝑙𝑔0.4 − 2𝑥𝑙𝑔0.5 = −3𝑙𝑔0.4
2 𝑙𝑜𝑔3 𝑅 − 𝑙𝑜𝑔3 𝑇 = 4 𝑥( 𝑙𝑔0.4− 2𝑙𝑔0.5) = −3(−0.3979)
𝑙𝑜𝑔3
𝑅2
𝑇
= 4 𝑥 ( −0.3979− 2(−0.3010)) = 3(0.3979)
𝑅2
𝑇
= 34
X= 5.8486
𝑅2
= 34
T
= 32
𝑇1/2
12. 𝑙𝑜𝑔75𝑥 = 4𝑙𝑜𝑔73 13.
2𝑙𝑜𝑔 𝑥3 +
1
2
𝑙𝑜𝑔 𝑥16 = 2
𝑙𝑜𝑔75𝑥 = 𝑙𝑜𝑔734 𝑙𝑜𝑔 𝑥32
+ 𝑙𝑜𝑔 𝑥161/2
= 2
5𝑥 = 34
𝑙𝑜𝑔 𝑥9 + 𝑙𝑜𝑔 𝑥4 = 2
𝑥 =
81
5
𝑙𝑜𝑔 𝑥9(4) = 2
𝑥 = 16
1
5
𝑙𝑜𝑔 𝑥36 = 2
36 = 𝑥2
62
= 𝑥2
X = 6

4. 14. 𝑙𝑔 5 + lg( 2𝑥 − 1) = lg 3 + lg( 𝑥 + 2)
lg 5(2𝑥 − 1) = lg 3(𝑥 + 2)
lg (10𝑥 − 5) = lg 3𝑥 + 6
( 10𝑥 − 5) = 3𝑥 + 6
10x-3x = 6+5
7x= 11
X=
11
7
15 4 𝑥
16
=
64
2 𝑦
𝑙𝑜𝑔 𝑥( 𝑦 + 2) = 1 + 𝑙𝑜𝑔 𝑥4
Find x and y
4 𝑥
16
=
64
2 𝑦
22𝑥
24 =
26
2 𝑦
22𝑥−4
=26−𝑦
2x-4=6-y
2x-4-6+y=0
2x+y= 10…..(1)
Solve simultaneous equation
𝑙𝑜𝑔 𝑥( 𝑦 + 2) = 1 + 𝑙𝑜𝑔 𝑥4 2x + y = 10
𝑙𝑜𝑔 𝑥( 𝑦 + 2) − 𝑙𝑜𝑔 𝑥4 = 1 (-) -4x + y = -2
𝑙𝑜𝑔 𝑥
𝑦 + 2
4
= 1
6x + 0 = 12
𝑦 + 2
4
= 𝑥1
6x = 12
Y+2= 4x x = 2
-4x+y=-2………….(2) Y+2=4x
Y+2=4(2)
Y= 6
16

5. 17 𝑙𝑜𝑔32 = 0.631 𝑙𝑜𝑔35 = 1.465
𝑙𝑜𝑔31.2 = 𝑙𝑜𝑔3
12
10
= 𝑙𝑜𝑔3
22
× 3
2 × 5
= 2𝑙𝑜𝑔32 + 𝑙𝑜𝑔33 − (𝑙𝑜𝑔32 + 𝑙𝑜𝑔35 )
= 2(0.631) +1 – (0.631 + 1.465)
=
18
𝑙𝑜𝑔3
25
4
= 𝑙𝑜𝑔3
52
22
=
2𝑙𝑜𝑔3 5
2𝑙𝑜𝑔3 2
=
2(1.465)
2(0.631)
=
19 𝑙𝑜𝑔5 √21 = 𝑙𝑜𝑔5 √3 × 7
= 𝑙𝑜𝑔53 × 71/2
1
2
𝑙𝑜𝑔5 ( 3 × 7)
1
2
(𝑙𝑜𝑔5 3 + 𝑙𝑜𝑔5 7)
=
20 2𝑙𝑜𝑔10(𝑥 − 𝑦) = 1 + 𝑙𝑜𝑔10 𝑥 + 𝑙𝑜𝑔10 𝑦
Proof 𝑥2
+ 𝑦2
= 12𝑥𝑦
𝑥2
+ 𝑦2
= 12𝑥𝑦
𝑙𝑜𝑔10(𝑥 − 𝑦)2
= 𝑙𝑜𝑔10 10 + 𝑙𝑜𝑔10 𝑥𝑦
𝑙𝑜𝑔10(𝑥 − 𝑦)2
− 𝑙𝑜𝑔1010 = 𝑙𝑜𝑔10 𝑥𝑦
(𝑥−𝑦)2
10
= xy
(𝑥 − 𝑦)2
= 10𝑥𝑦
( 𝑥 − 𝑦)( 𝑥 − 𝑦) = 10 xy
𝑥2
− 𝑥𝑦 − 𝑥𝑦 + 𝑦2
=10xy
𝑥2
+ 𝑦2
=12xy is proven