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CBSE Questions for Class 11 Commerce Applied Mathematics Logarithm And Antilogarithm Quiz 6 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Logarithm And Antilogarithm
Quiz 6
If the value of
(
log
10
2
)
+
1
is in the form of
log
10
m
, then
m
is equal to
Report Question
0%
11
0%
14
0%
20
0%
17
Explanation
log
10
2
+
1
=
log
10
2
+
log
10
10
=
log
10
(
2
×
10
)
=
log
10
20
The value of
log
10
0.001
is equal to
Report Question
0%
−
3
0%
3
0%
−
2
0%
2
Explanation
Consider,
log
10
0.001
=
x
⟹
0.001
=
10
x
⟹
1
1000
=
10
x
⟹
10
−
3
=
10
x
⟹
x
=
−
3
If
log
(
a
)
3
+
log
a
=
m
log
a
, then value of
m
is equal to
Report Question
0%
1
0%
4
0%
2
0%
−
5
Explanation
Step 1 : Use the rule of logarithm
According to logarithm power rule,
∵
log
e
(
X
Y
)
=
Y
.
log
e
(
X
)
Therefore,
log
(
a
)
3
=
3.
log
a
Since,
log
(
a
)
3
+
log
a
=
m
log
a
Therefore,
3.
log
a
+
log
a
=
m
log
a
4
log
a
=
m
log
a
By comparing both the sides we get,
m
=
4
Hence, option B is correct.
The logarithm form of
(
81
)
3
4
=
27
is
log
81
27
=
3
m
. Then value of
m
is equal to
Report Question
0%
2
0%
0
0%
4
0%
3
Explanation
Given,
log
81
27
=
3
m
...(1)
Now,
(
81
)
3
4
=
27
Converting in logarithm form, we can write
log
81
27
=
3
4
. ...(2)
From (1) and (2), we get
3
m
=
3
4
∴
m
=
4
If
log
81
log
27
=
x
, then the value of
x
is equal to
Report Question
0%
4
3
0%
3
4
0%
1
3
0%
Not solvable
Explanation
log
81
log
27
=
x
⇒
x
=
log
3
4
log
3
3
=
4
log
3
3
log
3
=
4
3
The logarithm form of
10
−
3
=
0.001
is
log
10
0.001
=
−
m
, then value of
m
is
Report Question
0%
−
1
0%
−
2
0%
3
0%
−
4
Explanation
Given,
10
−
3
=
0.001
⟹
log
10
0.001
=
−
3
∴
m
=
3
.
If
log
10
(
x
−
10
)
=
1
, then value of
x
is
Report Question
0%
10
0%
13
0%
20
0%
26
Explanation
Since,
log
10
(
x
−
10
)
=
1
⟹
(
x
−
10
)
=
10
1
⟹
x
=
20
.
If
64
a
=
1
256
b
, then
3
a
+
4
b
equals:
Report Question
0%
2
0%
4
0%
8
0%
0
Explanation
64
a
=
1
256
b
⇒
(
2
6
)
a
=
1
(
2
8
)
b
⇒
2
6
a
×
2
8
b
=
1
⇒
2
6
a
+
8
b
=
2
0
⇒
6
a
+
8
b
=
0
⇒
3
a
+
4
b
=
0
If
x
=
2
and
y
=
3
, then find the value of
[
1
x
x
+
1
y
y
]
.
Report Question
0%
−
31
108
0%
31
108
0%
125
171
0%
153
222
Explanation
1
x
x
+
1
y
y
=
1
2
2
+
1
3
3
=
1
4
+
1
27
=
27
+
4
108
=
31
108
If
log
4
m
=
1.5
, then the value of
m
is equal to
Report Question
0%
m
=
8
0%
m
=
4
0%
m
=
16
0%
m
=
64
Explanation
Since,
log
4
m
=
1.5
⇒
m
=
4
1.5
=
2
2
×
1.5
=
2
3
=
8
The value of
[
5
(
8
1
3
+
27
1
3
)
3
]
1
4
is
Report Question
0%
5
4
0%
5
1
4
0%
5
0%
None of these
Explanation
[
5
(
2
3
×
1
3
+
3
3
×
1
3
)
3
]
1
4
=
[
5
(
2
+
3
)
3
]
1
4
=
(
5
×
5
3
)
1
4
=
5
4
×
1
4
=
5
If
√
3
n
=
81
. Then n is equal to
Report Question
0%
2
0%
4
0%
6
0%
8
Explanation
⇒
√
3
n
=
81
⇒
3
n
/
2
=
3
4
Base are same so
⇒
n
2
=
4
⇒
n
=
8
The value of
512
−
2
9
is
Report Question
0%
1
2
0%
2
0%
4
0%
1
4
Explanation
(
512
)
−
2
9
=
(
2
9
)
−
2
9
=
2
9
×
−
2
9
∵
[
(
a
b
)
c
=
a
b
c
]
=
2
−
2
=
1
4
Hence, option
D
is correct.
The value of
(
−
3
)
0
−
(
−
3
)
3
−
(
−
3
)
−
1
+
(
−
3
)
4
−
(
−
3
)
−
2
is
Report Question
0%
109
2
9
0%
109
9
2
0%
109
0%
None of these
Explanation
Given expression is
(
−
3
)
0
−
(
−
3
)
3
−
(
−
3
)
−
1
+
(
−
3
)
4
−
(
−
3
)
−
2
Using the law of exponent,
a
−
m
=
1
a
m
, we can write it as:
=
1
−
(
−
27
)
−
(
−
1
3
)
+
81
−
1
9
=
1
+
27
+
1
3
+
81
−
1
9
=
109
+
2
9
=
109
2
9
Hence, option A is correct.
SImplify:
(
256
)
0.16
×
(
256
)
0.09
Report Question
0%
4
0%
16
0%
64
0%
256.25
Explanation
(
256
)
0.16
×
(
256
)
0.09
=
(
256
)
(
0.16
+
0.09
)
=
(
256
)
0.25
=
(
256
)
25
100
=
(
256
)
1
4
=
(
4
4
)
1
4
=
4
1
=
4
The value of
4
(
7
1
.
7
−
1
.
7
−
1
.7
0
)
is
Report Question
0%
3
7
3
0%
6
7
0%
3
3
7
0%
None of these
The value of
[
10
150
÷
10
146
]
is
Report Question
0%
1000
0%
10000
0%
100000
0%
10
5
Explanation
(
10
)
150
÷
(
10
)
146
=
(
10
)
150
(
10
)
146
=
10
(
150
−
146
)
=
10
4
=
10000
log
4
1 is equal to
Report Question
0%
1
0%
0
0%
∞
0%
none of these
Explanation
l
o
g
a
1
=
m
⇒
a
m
=
1
a
0
⇒
m
=
0
Value of
log
4
18
is:
Report Question
0%
an irrational number
0%
a rational number
0%
natural number
0%
whole number
Explanation
log
4
18
=
log
2
2
(
2.3
2
)
=
1
2
log
2
(
2.3
2
)
=
1
2
[
log
2
2
+
log
2
3
2
]
=
1
2
[
1
+
2.
log
2
3
]
log
2
3
is an irrational number.
Hence,
1
2
[
1
+
2.
log
2
3
]
is also an irrational number.
The value of
(
6
−
1
−
7
−
1
)
−
1
(
5
−
1
−
4
−
1
)
−
1
is equal to
Report Question
0%
1
2
0%
1
3
0%
1
4
0%
None of these
Explanation
Given expression is
(
6
−
1
−
7
−
1
)
−
1
(
5
−
1
−
4
−
1
)
−
1
.
From laws of exponents, we know that
a
−
m
=
(
1
a
)
m
Hence, the given expression can be simplified as:
(
1
6
−
1
7
)
−
1
(
1
5
−
1
4
)
−
1
Solving the fractions, we get:
1
6
−
1
7
=
1
42
and
1
5
−
1
4
=
−
1
20
Hence, the expression simplifies to:
(
1
42
)
−
1
(
−
1
20
)
−
1
Again applying the law of exponent,
a
−
m
=
(
1
a
)
m
, we get:
(
42
)
1
(
−
20
)
1
⇒
42
×
(
−
20
)
=
−
840
Hence,
(
6
−
1
−
7
−
1
)
−
1
(
5
−
1
−
4
−
1
)
−
1
=
−
840
.
Value of the expression
log
2
5
√
2.
3
√
2
√
2
is
Report Question
0%
0.1
0%
0.2
0%
0.3
0%
does not exist
Explanation
log
2
(
5
√
2.
3
√
2
√
2
)
=
=
log
2
(
5
√
2.
(
2
3
/
2
)
1
/
3
)
=
log
2
(
5
√
2.2
1
/
2
)
=
log
2
(
2
3
/
2
)
1
/
5
=
1
5
log
2
.
2
3
/
2
=
3
10
.
log
2
2
=
3
10
=
0.3
Simplify :
(
6
−
1
−
8
−
1
)
−
1
+
(
2
−
1
−
3
−
1
)
−
1
is-
Report Question
0%
20
0%
30
0%
10
0%
40
Explanation
Law of exponents,
Given -
(
6
−
1
−
8
−
1
)
−
1
+
(
2
−
1
−
3
−
1
)
−
1
=
(
1
6
−
1
8
)
−
1
+
(
1
2
−
1
3
)
−
1
=
(
8
−
6
6
×
8
)
−
1
+
(
3
−
2
2
×
3
)
−
1
=
(
2
36
×
8
)
−
1
+
(
1
2
×
3
)
−
1
=
(
1
24
)
−
1
+
(
1
6
)
−
1
=
24
+
6
=
30
Hence, option B is correct.
If
(
a
b
)
x
−
1
=
(
a
b
)
x
−
3
then the value of
x
is
Report Question
0%
-1
0%
1
0%
2
0%
3
Explanation
Given,
(
a
b
)
x
−
1
=
(
a
b
)
−
(
x
−
3
)
as the terms are same we can equate the powers/exponents of the terms
⇒
x
−
1
=
−
(
x
−
3
)
⇒
x
−
1
=
−
x
+
3
⇒
2
x
=
4
⇒
x
=
2
If
x
=
log
a
b
c
,
y
=
log
b
c
a
,
z
=
log
c
a
b
, then the value of
1
1
+
x
+
1
1
+
y
+
1
1
+
z
will be
Report Question
0%
x
+
y
+
z
0%
1
0%
a
b
+
b
c
+
c
a
0%
a
b
c
Explanation
Now,
1
+
x
=
log
a
a
+
log
a
b
c
=
log
a
a
b
c
⇒
1
1
+
x
=
log
a
b
c
a
Similarly,
1
1
+
y
=
log
a
b
c
b
and
1
1
+
z
=
log
a
b
c
c
∴
1
1
+
x
+
1
1
+
y
+
1
1
+
z
=
log
a
b
c
a
+
log
a
b
c
b
+
log
a
b
c
c
=
log
a
b
c
a
b
c
=
1
If
l
o
g
5
l
o
g
5
l
o
g
2
x
=
0
, then the value of
x
is
Report Question
0%
32
0%
125
0%
625
0%
25
Explanation
Given,
l
o
g
5
l
o
g
5
l
o
g
2
x
=
0
⇒
l
o
g
5
l
o
g
2
x
=
50
⇒
l
o
g
5
l
o
g
2
x
=
1
⇒
l
o
g
2
x
=
5
⇒
x
=
2
5
=
32
The value of
log
2
[
log
2
{
log
3
(
log
3
27
3
)
}
]
is equal to
Report Question
0%
2
0%
3
0%
0
0%
1
Explanation
Required value of
log
2
[
log
2
{
log
3
(
log
3
27
3
)
}
]
=
log
2
[
log
2
{
log
3
(
log
3
3
9
)
}
]
=
log
2
[
log
2
{
log
3
(
9
⋅
log
3
3
)
}
]
=
log
2
[
log
2
{
log
3
⋅
9
}
]
....
[
∵
log
3
3
=
1
]
=
log
2
[
log
2
{
log
3
3
2
}
]
=
log
2
[
log
2
2
⋅
log
3
3
]
=
log
2
[
log
2
2
]
....
[
∵
log
2
2
=
1
]
=
log
2
1
=
0
If there are
n
zeros after the decimal point, then the characteristic of that number will be
Report Question
0%
n
+
1
0%
−
n
+
1
0%
−
(
n
+
1
)
0%
n
−
1
Explanation
For number less than
1
,
if there are
n
zeroes after the decimal point, then the characteristics of that number will be
−
(
n
+
1
)
.
and For number greater than
1
,
if there are
n
number of zeroes are on the left sides of the digits then characteristics will be
(
n
+
1
)
.
Hence, C is the correct option.
The characteristic of a number having
m
(
m
>
1
)
digits is given by,
Report Question
0%
m
−
1
0%
m
+
1
0%
m
0%
None of the above
Explanation
If a number is
N
>
0
, t
hen
log
10
N
will have two parts, the integral part is known the characteristic and the decimal part is known as mantissa.
2
digit belongs from
[
10
,
100
]
where
log
10
10
=
1
and
log
10
100
=
2
Similarly,
m
digit number belongs from
[
10
m
−
1
,
10
m
]
, where
log
10
10
m
−
1
=
m
−
1
and
log
10
10
m
=
m
.
Thus any number between
[
10
m
−
1
,
10
m
]
will have
m
−
1
as the integral part.
Thus the characteristic of a number having
m
digits is given by
m
−
1
.
Let
x
=
(
0.15
)
20
. Find the characteristic in the logarithm of
x
to the base
10
.
Report Question
0%
17
0%
21
0%
−
21
0%
−
17
Explanation
Given
x
=
(
0.15
)
20
By applying
log
on both sides , we get
log
10
x
=
20
log
10
(
0.15
)
=
−
16.478
⇒
log
10
x
=
−
17
+
0.5218
The integral part of
log
10
x
is called characteristic
Therefore the characteristic of given number is
−
17
So option
D
is correct
The value of
log
10
0.0006024
is equal to
Report Question
0%
¯
3
.7979
0%
¯
1
.9779
0%
¯
4
.7799
0%
0.7279
Explanation
log
10
0.0006024
Characteristics
=
−
4
For mantissa ,read from the table
6024.
Mantissa
=
7799
Thus,
log
10
0.0006024
=
characteristics of
0.0006024
+
mantissa of
0.0006024
=
−
4
+
0.7799
=
ˉ
4
.7799
Hence, C is the correct option.
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