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Page 1
JEE Mains Previous Year Questions
(2021-2024): Logarithm
2023
Q1: Let ?? , ?? , ?? be three distinct positive real numbers such that ( ?? ?? )
?? ?? ?? ?? ? ?? = ( ???? )
?? ?? ?? ?? ? ?? and ?? ?? ?? ?? ?? ? ?? =
?? ?? ?? ?? ?? ? ?? .
Then, ?? ?? + ?? ???? is equal to [JEE Main 2023 (Online) 10th April Morning Shift]
Ans: 8
Given, ( 2 ?? )
ln ? ?? = ( ???? )
ln ? ?? , where 2 ?? > 0 , ???? > 0
? ln ? ?? ( ln ? 2 + ln ? ?? ) = ln ? ?? ( ln ? ?? + ln ? ?? ) .
and ( ?? )
ln ? 2
= ( ?? )
ln ? ??
? ln ? 2 · ln ? ?? = ln ? ?? · ln ? ?? .
Now, let ln ? ?? = ?? , ln ? ?? = ??
ln ? 2 = ?? , ln ? ?? = ??
Now, from Eqs. (i) and (ii), we get
?? · ?? = ???? and ?? ( ?? + ?? ) = ?? ( ?? + ?? )
? ?? =
????
??
? ? ?? (
????
?? + ?? ) = ?? ( ?? + ?? )
? ? ?? 2
?? + ?? 2
?? = ?? 2
( ?? + ?? )
? ? ( ?? 2
- ?? 2
) ( ?? + ?? ) = 0
? ? ?? 2
= ?? 2
?? = ± ??
So, from the equations, there are two cases:
Case ?? :
?? = ??
In this case, since ?? and ?? are natural logarithms of positive numbers ?? and ?? respectively, this implies
that ?? = ?? . However, this cannot be true as a, ?? , and ?? are given to be distinct positive real numbers.
Page 2
JEE Mains Previous Year Questions
(2021-2024): Logarithm
2023
Q1: Let ?? , ?? , ?? be three distinct positive real numbers such that ( ?? ?? )
?? ?? ?? ?? ? ?? = ( ???? )
?? ?? ?? ?? ? ?? and ?? ?? ?? ?? ?? ? ?? =
?? ?? ?? ?? ?? ? ?? .
Then, ?? ?? + ?? ???? is equal to [JEE Main 2023 (Online) 10th April Morning Shift]
Ans: 8
Given, ( 2 ?? )
ln ? ?? = ( ???? )
ln ? ?? , where 2 ?? > 0 , ???? > 0
? ln ? ?? ( ln ? 2 + ln ? ?? ) = ln ? ?? ( ln ? ?? + ln ? ?? ) .
and ( ?? )
ln ? 2
= ( ?? )
ln ? ??
? ln ? 2 · ln ? ?? = ln ? ?? · ln ? ?? .
Now, let ln ? ?? = ?? , ln ? ?? = ??
ln ? 2 = ?? , ln ? ?? = ??
Now, from Eqs. (i) and (ii), we get
?? · ?? = ???? and ?? ( ?? + ?? ) = ?? ( ?? + ?? )
? ?? =
????
??
? ? ?? (
????
?? + ?? ) = ?? ( ?? + ?? )
? ? ?? 2
?? + ?? 2
?? = ?? 2
( ?? + ?? )
? ? ( ?? 2
- ?? 2
) ( ?? + ?? ) = 0
? ? ?? 2
= ?? 2
?? = ± ??
So, from the equations, there are two cases:
Case ?? :
?? = ??
In this case, since ?? and ?? are natural logarithms of positive numbers ?? and ?? respectively, this implies
that ?? = ?? . However, this cannot be true as a, ?? , and ?? are given to be distinct positive real numbers.
Case 2 :
?? = - ??
In this case, ln ? ?? = - ln ? ??
? ?? × ?? = 1
? ?? =
1
??
Also, ?? + ?? = 0 (from the equations above)
? ? ln ? ?? + ln ? ?? = 0
? ? ln ? ( ?? × ?? ) = 0
? ? ?? × ?? = 1
Given ?? =
1
?? and ?? × ?? = 1 ? ?? = ??
Thus, in the case where ?? = - ?? , the possible values are:
?? =
1
?? ?? = ??
If ???? = 1 ? ( 2 ?? )
ln ? ?? = 1
? ?? = 1 / 2
So, 6 ?? + 5 ???? = 6 (
1
2
) + 5 = 3 + 5 = 8
Q2: Let ?? = { ?? : ??????
?? ? ( ?? ?? ?? - ?? + ???? ) - ??????
?? ? (
?? ?? · ?? ?? ?? - ?? + ?? ) = ?? }. Then the maximum value of ?? for
which the equation ?? ?? - ?? ( ?
?? ? ?? ? ?? )
?? ?? + ?
?? ? ?? ? ( ?? + ?? )
?? ?? = ?? has real roots, is [JEE Main 2023
(Online) 25th January Morning Shift]
Ans: 25
Page 3
JEE Mains Previous Year Questions
(2021-2024): Logarithm
2023
Q1: Let ?? , ?? , ?? be three distinct positive real numbers such that ( ?? ?? )
?? ?? ?? ?? ? ?? = ( ???? )
?? ?? ?? ?? ? ?? and ?? ?? ?? ?? ?? ? ?? =
?? ?? ?? ?? ?? ? ?? .
Then, ?? ?? + ?? ???? is equal to [JEE Main 2023 (Online) 10th April Morning Shift]
Ans: 8
Given, ( 2 ?? )
ln ? ?? = ( ???? )
ln ? ?? , where 2 ?? > 0 , ???? > 0
? ln ? ?? ( ln ? 2 + ln ? ?? ) = ln ? ?? ( ln ? ?? + ln ? ?? ) .
and ( ?? )
ln ? 2
= ( ?? )
ln ? ??
? ln ? 2 · ln ? ?? = ln ? ?? · ln ? ?? .
Now, let ln ? ?? = ?? , ln ? ?? = ??
ln ? 2 = ?? , ln ? ?? = ??
Now, from Eqs. (i) and (ii), we get
?? · ?? = ???? and ?? ( ?? + ?? ) = ?? ( ?? + ?? )
? ?? =
????
??
? ? ?? (
????
?? + ?? ) = ?? ( ?? + ?? )
? ? ?? 2
?? + ?? 2
?? = ?? 2
( ?? + ?? )
? ? ( ?? 2
- ?? 2
) ( ?? + ?? ) = 0
? ? ?? 2
= ?? 2
?? = ± ??
So, from the equations, there are two cases:
Case ?? :
?? = ??
In this case, since ?? and ?? are natural logarithms of positive numbers ?? and ?? respectively, this implies
that ?? = ?? . However, this cannot be true as a, ?? , and ?? are given to be distinct positive real numbers.
Case 2 :
?? = - ??
In this case, ln ? ?? = - ln ? ??
? ?? × ?? = 1
? ?? =
1
??
Also, ?? + ?? = 0 (from the equations above)
? ? ln ? ?? + ln ? ?? = 0
? ? ln ? ( ?? × ?? ) = 0
? ? ?? × ?? = 1
Given ?? =
1
?? and ?? × ?? = 1 ? ?? = ??
Thus, in the case where ?? = - ?? , the possible values are:
?? =
1
?? ?? = ??
If ???? = 1 ? ( 2 ?? )
ln ? ?? = 1
? ?? = 1 / 2
So, 6 ?? + 5 ???? = 6 (
1
2
) + 5 = 3 + 5 = 8
Q2: Let ?? = { ?? : ??????
?? ? ( ?? ?? ?? - ?? + ???? ) - ??????
?? ? (
?? ?? · ?? ?? ?? - ?? + ?? ) = ?? }. Then the maximum value of ?? for
which the equation ?? ?? - ?? ( ?
?? ? ?? ? ?? )
?? ?? + ?
?? ? ?? ? ( ?? + ?? )
?? ?? = ?? has real roots, is [JEE Main 2023
(Online) 25th January Morning Shift]
Ans: 25
log
2
? ( 9
2 ?? - 4
+ 13 ) - log
2
? (
5
2
· 3
2 ?? - 4
+ 1 ) = 2
? ?
9
2 ?? - 4
+ 13
5
2
3
2 ?? - 4
+ 1
= 4
? ? ?? = 2 or 3
? ? ?
?? ? S
? ?? = 5 and ? ?
?? ? S
? ( ?? + 1 )
2
= 25
? ? ?? 2
- 50 ?? + 25 ?? = 0 has real roots
? ? ?? = 25
? ? ?? m a x
= 25
Q3: The number of integral solutions ?? of ??????
( ?? +
?? ?? )
? (
?? - ?? ?? ?? - ?? )
?? = ?? is:
A. 8
B. 7
C. 5
D. 6 [JEE Main 2023 (Online) 11th April Morning Shift]
Ans: ( d )
log
( ?? +
7
2
)
? (
?? - 7
2 ?? - 3
)
2
= 0
Domain :
?? +
7
2
> 0
?? >
- 7
2
?? +
7
2
? 1
?? ?
- 5
2
?? - 7
2 ?? - 3
? 0
?? ? 7
?? ?
3
2
Taking intersection : ?? ? (
- 7
2
, 8 ) - { -
5
2
,
3
2
, 7 }
Now log
a
? b = 0 if a > 1 and b = 1
Page 4
JEE Mains Previous Year Questions
(2021-2024): Logarithm
2023
Q1: Let ?? , ?? , ?? be three distinct positive real numbers such that ( ?? ?? )
?? ?? ?? ?? ? ?? = ( ???? )
?? ?? ?? ?? ? ?? and ?? ?? ?? ?? ?? ? ?? =
?? ?? ?? ?? ?? ? ?? .
Then, ?? ?? + ?? ???? is equal to [JEE Main 2023 (Online) 10th April Morning Shift]
Ans: 8
Given, ( 2 ?? )
ln ? ?? = ( ???? )
ln ? ?? , where 2 ?? > 0 , ???? > 0
? ln ? ?? ( ln ? 2 + ln ? ?? ) = ln ? ?? ( ln ? ?? + ln ? ?? ) .
and ( ?? )
ln ? 2
= ( ?? )
ln ? ??
? ln ? 2 · ln ? ?? = ln ? ?? · ln ? ?? .
Now, let ln ? ?? = ?? , ln ? ?? = ??
ln ? 2 = ?? , ln ? ?? = ??
Now, from Eqs. (i) and (ii), we get
?? · ?? = ???? and ?? ( ?? + ?? ) = ?? ( ?? + ?? )
? ?? =
????
??
? ? ?? (
????
?? + ?? ) = ?? ( ?? + ?? )
? ? ?? 2
?? + ?? 2
?? = ?? 2
( ?? + ?? )
? ? ( ?? 2
- ?? 2
) ( ?? + ?? ) = 0
? ? ?? 2
= ?? 2
?? = ± ??
So, from the equations, there are two cases:
Case ?? :
?? = ??
In this case, since ?? and ?? are natural logarithms of positive numbers ?? and ?? respectively, this implies
that ?? = ?? . However, this cannot be true as a, ?? , and ?? are given to be distinct positive real numbers.
Case 2 :
?? = - ??
In this case, ln ? ?? = - ln ? ??
? ?? × ?? = 1
? ?? =
1
??
Also, ?? + ?? = 0 (from the equations above)
? ? ln ? ?? + ln ? ?? = 0
? ? ln ? ( ?? × ?? ) = 0
? ? ?? × ?? = 1
Given ?? =
1
?? and ?? × ?? = 1 ? ?? = ??
Thus, in the case where ?? = - ?? , the possible values are:
?? =
1
?? ?? = ??
If ???? = 1 ? ( 2 ?? )
ln ? ?? = 1
? ?? = 1 / 2
So, 6 ?? + 5 ???? = 6 (
1
2
) + 5 = 3 + 5 = 8
Q2: Let ?? = { ?? : ??????
?? ? ( ?? ?? ?? - ?? + ???? ) - ??????
?? ? (
?? ?? · ?? ?? ?? - ?? + ?? ) = ?? }. Then the maximum value of ?? for
which the equation ?? ?? - ?? ( ?
?? ? ?? ? ?? )
?? ?? + ?
?? ? ?? ? ( ?? + ?? )
?? ?? = ?? has real roots, is [JEE Main 2023
(Online) 25th January Morning Shift]
Ans: 25
log
2
? ( 9
2 ?? - 4
+ 13 ) - log
2
? (
5
2
· 3
2 ?? - 4
+ 1 ) = 2
? ?
9
2 ?? - 4
+ 13
5
2
3
2 ?? - 4
+ 1
= 4
? ? ?? = 2 or 3
? ? ?
?? ? S
? ?? = 5 and ? ?
?? ? S
? ( ?? + 1 )
2
= 25
? ? ?? 2
- 50 ?? + 25 ?? = 0 has real roots
? ? ?? = 25
? ? ?? m a x
= 25
Q3: The number of integral solutions ?? of ??????
( ?? +
?? ?? )
? (
?? - ?? ?? ?? - ?? )
?? = ?? is:
A. 8
B. 7
C. 5
D. 6 [JEE Main 2023 (Online) 11th April Morning Shift]
Ans: ( d )
log
( ?? +
7
2
)
? (
?? - 7
2 ?? - 3
)
2
= 0
Domain :
?? +
7
2
> 0
?? >
- 7
2
?? +
7
2
? 1
?? ?
- 5
2
?? - 7
2 ?? - 3
? 0
?? ? 7
?? ?
3
2
Taking intersection : ?? ? (
- 7
2
, 8 ) - { -
5
2
,
3
2
, 7 }
Now log
a
? b = 0 if a > 1 and b = 1
Or
?? ? ( 0 , 1 ) and ?? ? ( 0 , 1 )
Case I: ?? +
7
2
> 1 and (
?? - 7
2 ?? - 3
)
2
= 1
? ?? > -
5
2
and
? ( 2 ?? - 3 )
2
- ( ?? - 7 )
2
= 0
? ( 2 ?? - 3 + ?? - 7 ) ( 2 ?? - 3 - ?? + 7 ) = 0
? ( 3 ?? - 10 ) ( ?? + 4 ) = 0
?? ? [ - 4 ,
10
3
]
Intersection : x ? (
- 5
2
,
10
3
]
Intersection : ?? ? (
- 5
2
,
10
3
]
Case II: ?? +
7
2
? ( 0 , 1 ) and (
?? - 7
2 ?? - 3
)
2
? ( 0 , 1 )
? ? 0 < ?? +
7
2
< 1
? -
7
2
< ?? <
- 5
2
and
(
?? - 7
2 ?? - 3
)
2
< 1
? ? ( ?? - 7 )
2
< ( 2 ?? - 3 )
2
? ?? ? ( - 8 , - 4 ) ? (
10
3
, 8 )
No common values of x.
Hence intersection with feasible region.
We get ?? ? (
- 5
2
,
10
3
] - {
3
2
}
Integral value of ?? are { - 2 , - 1 , 0 , 1 , 2 , 3 }
No. of integral values = 6
Page 5
JEE Mains Previous Year Questions
(2021-2024): Logarithm
2023
Q1: Let ?? , ?? , ?? be three distinct positive real numbers such that ( ?? ?? )
?? ?? ?? ?? ? ?? = ( ???? )
?? ?? ?? ?? ? ?? and ?? ?? ?? ?? ?? ? ?? =
?? ?? ?? ?? ?? ? ?? .
Then, ?? ?? + ?? ???? is equal to [JEE Main 2023 (Online) 10th April Morning Shift]
Ans: 8
Given, ( 2 ?? )
ln ? ?? = ( ???? )
ln ? ?? , where 2 ?? > 0 , ???? > 0
? ln ? ?? ( ln ? 2 + ln ? ?? ) = ln ? ?? ( ln ? ?? + ln ? ?? ) .
and ( ?? )
ln ? 2
= ( ?? )
ln ? ??
? ln ? 2 · ln ? ?? = ln ? ?? · ln ? ?? .
Now, let ln ? ?? = ?? , ln ? ?? = ??
ln ? 2 = ?? , ln ? ?? = ??
Now, from Eqs. (i) and (ii), we get
?? · ?? = ???? and ?? ( ?? + ?? ) = ?? ( ?? + ?? )
? ?? =
????
??
? ? ?? (
????
?? + ?? ) = ?? ( ?? + ?? )
? ? ?? 2
?? + ?? 2
?? = ?? 2
( ?? + ?? )
? ? ( ?? 2
- ?? 2
) ( ?? + ?? ) = 0
? ? ?? 2
= ?? 2
?? = ± ??
So, from the equations, there are two cases:
Case ?? :
?? = ??
In this case, since ?? and ?? are natural logarithms of positive numbers ?? and ?? respectively, this implies
that ?? = ?? . However, this cannot be true as a, ?? , and ?? are given to be distinct positive real numbers.
Case 2 :
?? = - ??
In this case, ln ? ?? = - ln ? ??
? ?? × ?? = 1
? ?? =
1
??
Also, ?? + ?? = 0 (from the equations above)
? ? ln ? ?? + ln ? ?? = 0
? ? ln ? ( ?? × ?? ) = 0
? ? ?? × ?? = 1
Given ?? =
1
?? and ?? × ?? = 1 ? ?? = ??
Thus, in the case where ?? = - ?? , the possible values are:
?? =
1
?? ?? = ??
If ???? = 1 ? ( 2 ?? )
ln ? ?? = 1
? ?? = 1 / 2
So, 6 ?? + 5 ???? = 6 (
1
2
) + 5 = 3 + 5 = 8
Q2: Let ?? = { ?? : ??????
?? ? ( ?? ?? ?? - ?? + ???? ) - ??????
?? ? (
?? ?? · ?? ?? ?? - ?? + ?? ) = ?? }. Then the maximum value of ?? for
which the equation ?? ?? - ?? ( ?
?? ? ?? ? ?? )
?? ?? + ?
?? ? ?? ? ( ?? + ?? )
?? ?? = ?? has real roots, is [JEE Main 2023
(Online) 25th January Morning Shift]
Ans: 25
log
2
? ( 9
2 ?? - 4
+ 13 ) - log
2
? (
5
2
· 3
2 ?? - 4
+ 1 ) = 2
? ?
9
2 ?? - 4
+ 13
5
2
3
2 ?? - 4
+ 1
= 4
? ? ?? = 2 or 3
? ? ?
?? ? S
? ?? = 5 and ? ?
?? ? S
? ( ?? + 1 )
2
= 25
? ? ?? 2
- 50 ?? + 25 ?? = 0 has real roots
? ? ?? = 25
? ? ?? m a x
= 25
Q3: The number of integral solutions ?? of ??????
( ?? +
?? ?? )
? (
?? - ?? ?? ?? - ?? )
?? = ?? is:
A. 8
B. 7
C. 5
D. 6 [JEE Main 2023 (Online) 11th April Morning Shift]
Ans: ( d )
log
( ?? +
7
2
)
? (
?? - 7
2 ?? - 3
)
2
= 0
Domain :
?? +
7
2
> 0
?? >
- 7
2
?? +
7
2
? 1
?? ?
- 5
2
?? - 7
2 ?? - 3
? 0
?? ? 7
?? ?
3
2
Taking intersection : ?? ? (
- 7
2
, 8 ) - { -
5
2
,
3
2
, 7 }
Now log
a
? b = 0 if a > 1 and b = 1
Or
?? ? ( 0 , 1 ) and ?? ? ( 0 , 1 )
Case I: ?? +
7
2
> 1 and (
?? - 7
2 ?? - 3
)
2
= 1
? ?? > -
5
2
and
? ( 2 ?? - 3 )
2
- ( ?? - 7 )
2
= 0
? ( 2 ?? - 3 + ?? - 7 ) ( 2 ?? - 3 - ?? + 7 ) = 0
? ( 3 ?? - 10 ) ( ?? + 4 ) = 0
?? ? [ - 4 ,
10
3
]
Intersection : x ? (
- 5
2
,
10
3
]
Intersection : ?? ? (
- 5
2
,
10
3
]
Case II: ?? +
7
2
? ( 0 , 1 ) and (
?? - 7
2 ?? - 3
)
2
? ( 0 , 1 )
? ? 0 < ?? +
7
2
< 1
? -
7
2
< ?? <
- 5
2
and
(
?? - 7
2 ?? - 3
)
2
< 1
? ? ( ?? - 7 )
2
< ( 2 ?? - 3 )
2
? ?? ? ( - 8 , - 4 ) ? (
10
3
, 8 )
No common values of x.
Hence intersection with feasible region.
We get ?? ? (
- 5
2
,
10
3
] - {
3
2
}
Integral value of ?? are { - 2 , - 1 , 0 , 1 , 2 , 3 }
No. of integral values = 6
Q4: If the solution of the equation ??????
???? ?? ? ?? ? ?? ???? ? ?? + ?? ??????
?? ?? ?? ? ?? ? ?? ?? ?? ? ?? = ?? , ?? ? ( ?? ,
?? ?? ), is ?? ???? - ?? ? (
?? + v ?? ?? ),
where ?? , ?? are integers, then ?? + ?? is equal to:
A. 3
B. 6
C. 4
D. 5 [JEE Main 2023 (Online) 30th January Morning Shift]
Ans: (c)
log
c o s ? ?? ? c o t ? ?? + 4 log
s i n ? ?? ? t a n ? ?? = 1
? ? log
c o s ? ?? ? c o t ? ?? - 4 log
s i n ? ?? ? c o t ? ?? = 1
? ? 1 - log
c o s ? ?? ? sin ? ?? - 4 - 4 log
s i n ? ?? ? c o s ? ?? = 1
Let log
c o s ? ?? ? sin ? ?? = ?? ?? +
4
?? = 4
? ? ?? = 2
sin ? ?? = c o s
2
? ?? ? ? sin ? ?? = 1 - sin
2
? ?? ? ? sin
2
? ?? + sin ? ?? - 1
= 0
? ? sin ? ?? =
- 1 ± v 5
2
as ?? ? ( 0 ,
?? 2
)
sin ? ?? =
v 5 - 1
2
?? = sin
- 1
? (
- 1 + v 5
2
)
? ? ?? = - 1 , ?? = 5
?? + ?? = 4
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FAQs on Logarithm: JEE Main Previous Year Questions (2021-2025) - Mathematics (Maths) for JEE Main & Advanced
| 1. What is the basic concept of logarithm? |  |
Ans. Logarithm is the inverse operation of exponentiation. It helps in solving problems involving exponential functions by transforming them into simpler forms using logarithmic properties.
| 2. How is the logarithm of a number calculated? | |
Ans. The logarithm of a number is calculated by taking the exponent to which a base number must be raised to obtain the given number. For example, log₂8 = 3 because 2³ = 8.
| 3. What are the properties of logarithms? | |
Ans. Some key properties of logarithms include the product rule (logₐ(MN) = logₐM + logₐN), quotient rule (logₐ(M/N) = logₐM - logₐN), and power rule (logₐMᴺ = N*logₐM). These properties help simplify logarithmic expressions.
| 4. How are logarithms used in real-life applications? | |
Ans. Logarithms are used in various real-life applications such as earthquake magnitude scales (Richter scale), measuring sound intensity (decibels), and calculating pH levels in chemistry. They help compress large ranges of values into more manageable scales.
| 5. What are common mistakes to avoid when working with logarithms? | |
Ans. Common mistakes when working with logarithms include forgetting to apply logarithmic properties correctly, misinterpreting the base of the logarithm, and not simplifying expressions before solving. It is important to pay attention to detail and review basic logarithmic rules to avoid errors.
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Logarithm: JEE Main Previous Year Questions (2021-2025) JEE
The "Logarithm: JEE Main Previous Year Questions (2021-2025) JEE Questions" guide is a valuable resource for all aspiring students preparing for the
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The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
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The content of Logarithm: JEE Main Previous Year Questions (2021-2025) is prepared as per the latest JEE syllabus.
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