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1. JEE MAIN APRIL 2021 : MATH 30 Qs. in 45 mins.
WORK
4 U!
M
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C
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N
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H
A
A
G
R
A
W
A
L

2. N
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H
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WORK
4 U!
Let the MAGIC BEGIN !!

3. N
E
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A
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A
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WORK
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4. N
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A
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D
WORK
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5. N
E
H
A
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A
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I
N
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D
WORK
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6. N
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A
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WORK
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7. N
E
H
A
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M
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I
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A
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D
WORK
4 U!
Let the MAGIC BEGIN !!

8. FOUNDATION SERIES MATH : LOGARITHMS
WORK
4 U!
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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
WORK
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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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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
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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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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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log10
30 + log10
20= x
?

25. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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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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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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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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Property-9

38. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Property-10

39. N
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WORK
4 U!
Let the MAGIC BEGIN !!

40. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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D

41. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Q1. Find the value of log2

42. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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=
=
=
2
3
log2
2
=
2
3
Solution:

43. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Q2. Find the value of log 0.01
1000

44. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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=
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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Q3. If then what is x?

46. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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Solution:

47. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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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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Solution:

49. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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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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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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Solution:

52. FOUNDATION SERIES MATH : LOGARITHMS
WORK
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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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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
4
8
64
16
A
B
D
C
Q6. If xy2
= 4 and
then x equals

54. 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:

55. 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
4
8
64
16
Q6. If xy2
= 4 and
then x equals
A
B
D
C

56. 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
WORK
4 U!
Let the MAGIC BEGIN !!

57. 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
Q7.

58. 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
Q7.
Answer : 0

59. 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
Q8.

60. 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
Q8.
Answer : 0

61. 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
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 :