8. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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We know that
100 = 102
1000 = 103
What if you are asked Find ‘x’, if 39 = 10x
?
9. FOUNDATION SERIES MATH : LOGARITHMS
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x = log10
39
10y
= x ⇔ y = log 10
x
Base of
logarithm
The ‘x’ value in above question is
10. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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loga
x = y
log2
32 = k
⇔ x = ay
Example
11. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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loga
x = y
log2
32 = k
⇒ 32 = 2k
⇒ 25
= 2k
⇒ k = 5
⇔ x = ay
Example
12. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Note
Common Logarithms:
Logarithm with base ‘10’. In all science
applications the base is taken as ‘10’.
Natural Logarithms:
The logarithms with base ‘e’
(e = 2.718. . ) which is also an irrational
number like ‘π’
1)
2)
loge
x = ln x
13. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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1) x > 0
2) a > 0
3) a ≠ 1
y = loga
x
The necessary conditions for to
exist are
loga
x
14. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 1
(a ≠ 1, a > 0)
log a
1 = 0
15. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 1
log a
1 = 0
(a ≠ 1, a > 0)
1 = ak
a0
= ak
⇒ k = 0
Proof
log a
1 = k
Let
log a
1 = 0
Logarithm of ‘1’ is always zero
16. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
log4
1 = y
?
17. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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log4
1 = y
?
Answer : y = 0
18. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 2
log a
a = 1 (a ≠ 1, a > 0)
19. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
log20
20 = y
?
20. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 2
log a
a = 1 (a ≠ 1, a > 0)
Let log a
a = k
⇒ a = ak
⇒ k = 1
Example
log2
2 = log3
3 =1
log0.1
0.1 = 1
Proof
21. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Property - 3
loga
x1
+ loga
x2
= loga
(x1
x2
)
22. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 3
loga
x1
+ loga
x2
= loga
(x1
x2
)
Proof
23. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property - 3
loga
x1
+ loga
x2
= loga
(x1
x2
)
Let loga
x1
= k1
⇒ x1
x2
= ak1 + k2
loga
(x1
x2
) = k1
+ k2
loga
x2
= k2
x1
= ak1
x2
= ak2
= loga
x1
+loga
x2
&
&
Proof
24. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
log10
30 + log10
20= x
?
25. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Property - 4
loga
x1
– loga
x2
=
loga
x1
x2
26. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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log10
30 - log10
20= x
?
27. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Property 5
log a
(xn
) = n log a
x (x > 0, a > 0, a ≠ 1)
28. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
log10
100 2
= x
?
29. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property 5
log a
(xn
) = n log a
x (x > 0, a > 0, a ≠ 1)
Let loga
(xn
) = k
xn
= ak
x = a(k/n)
k
n
= loga
x
k = n loga
x
Proof
30. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property 6
31. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property 6
Proof
32. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property-7
a log b
= b log
a
33. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property-7
a log b
= b log
a
a log b
=
k1
⇒ log b = loga
k1
b log a
= k2
⇒ log a = logb
k2
log b =
log k1
log a
⇒ log k1
= (log a) (logb)
⇒ log k2
(log a) (logb)
log a =
log k2
log b
log k1
log k2
=
⇒ k1
k2
=
Proof
=
⇒
34. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property-8
35. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Q. Find
36. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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L.H.S = (x)log
y
z (y)log
z
x (z) log
x
y
= x
log y – log z
y
log z – log x
z
log x – log y
x
log y
x
log z
=
y
log z
y
log x
×
z
log x
z
log y
×
= 1
Solution :
37. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Property-9
38. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Property-10
40. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
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D
41. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Q1. Find the value of log2
42. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
=
=
=
2
3
log2
2
=
2
3
Solution:
43. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Q2. Find the value of log 0.01
1000
44. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
=
log 0.01
1000 log 1
100
103
= 3 log
10
–2
10
=
3
–2
log10
10
=
– 3
2
Solution:
45. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Q3. If then what is x?
46. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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C
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D
Solution:
47. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Q4. If ,
then,
2a = b
A
B
D
C
a = 2b
a = b
2a = 3b
48. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D
Solution:
49. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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C
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I
N
E
D
Q4. If ,
then,
2a = b
A
B
D
C
a = 2b
a = b
2a = 3b
50. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
N
C
L
I
N
E
D
3( log a + log b + log c)
3 log a
0
-3(log b + log c)
Q5.
is equal to
A
B
D
C
51. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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A
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I
N
C
L
I
N
E
D
Solution:
52. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
N
C
L
I
N
E
D
3( log a + log b + log c)
3 log a
0
-3(log b + log c)
Q5.
is equal to
A
B
D
C
53. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
N
C
L
I
N
E
D
4
8
64
16
A
B
D
C
Q6. If xy2
= 4 and
then x equals
54. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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C
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D
Solution:
55. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
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C
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I
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E
D
4
8
64
16
Q6. If xy2
= 4 and
then x equals
A
B
D
C
57. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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A
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Y
I
N
C
L
I
N
E
D
Q7.
58. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
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C
L
I
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E
D
Q7.
Answer : 0
59. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
N
C
L
I
N
E
D
Q8.
60. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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I
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C
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D
Q8.
Answer : 0
61. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
Q9. find M.
62. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
Answer : M = 24
Q9. find M.
63. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
A
B
D
C
Q10. If a = log2, b = log3, c = log7
6x
= 7x+4
, then x =
64. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
A
B
D
C
Q10. If a = log2, b = log3, c = log7
6x
= 7x+4
, then x =
65. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
A
B
D
C
Q11. If n=(2017)! then
is
0
1
n
66. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
A
B
D
C
Q11. If n=(2017)! then
is
0
1
n
67. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
Q12. Solve
68. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
N
E
H
A
A
G
R
A
W
A
L
M
A
T
H
E
M
A
T
I
C
A
L
L
Y
I
N
C
L
I
N
E
D
Solution :
