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CAT Questions | CAT Arithmetic

CAT Quantitative Aptitude | CAT Exponents and Logarithms

CAT Exponents and Logarithms Questions are one of the most commonly tested topics in CAT exam. Questions from Exponents and Logarithms have appeared consistently in the CAT exam for the last several years. Questions from Exponents and Logarithms range from very easy to very hard. The basic concept is very easy, learn the concepts and practice a wide range of CAT Questions from 2IIM. One can usually expect 2-3 questions from Logarithms and Exponents in the CAT exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

CAT Exponents and Logarithms: Inequalities

If log₂X + log₄X = log_0.25√6 and x > 0, then x is
1. 6^-1/6
2. 6^1/6
3. 3^-1/3
4. 6^1/3
Choice A
6^-1/6
CAT Exponents and Logarithms: Simple

log₉ (3log₂ (1 + log₃ (1 + 2log₂x))) = \\frac{1}{2}\\). Find x.
1. 4
2. \\frac{1}{2}\\)
3. 1
4. 2
Choice D
2
CAT Exponents and Logarithms: Quadratic Equations

If 2^2x+4 – 17 × 2^x+1 = –4, then which of the following is true?
1. x is a positive value
2. x is a negative value
3. x can be either a positive value or a negative value
4. None of these
Choice C
x can be either a positive value or a negative value
CAT Exponents and Logarithms: Algebra

If log₁₂27 = a, log₉16 = b, find log₈108.
1. \\frac{2(a + 3)}{3b}\\)
2. \\frac{2(a + 3)}{3a}\\)
3. \\frac{2(b + 3)}{3a}\\)
4. \\frac{2(b + 3)}{3b}\\)
Choice D
\\frac{2(b + 3)}{3b}\\)
CAT Exponents and Logarithms: Inequalities

\\frac{log_3(x-3)}{log_3(x-5)}\\) < 0. If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b.
1. 214
2. 216
3. 200
4. 203
Choice A
214
CAT Exponents and Logarithms: Different bases

log₅x = a (This should be read as log X to the base 5 equals a) log₂₀x = b. What is log_x10?
1. \\frac{a + b}{2ab}\\)
2. (a + b) * 2ab
3. \\frac{2ab}{a + b}\\)
4. \\frac{a + b}{2}\\)
Choice A
\\frac{a + b}{2ab}\\)
CAT Exponents and Logarithms: Basic identities of Logarithm

log₃x + log_x3 = \\frac{17}{4}\\). Find the value of x.
1. 3⁴
2. 3^1/4
3. 3⁴ or 3^1/4
4. 3^1/3
Choice C
3⁴ or 3^1/4
CAT Exponents and Logarithms: Basic identities of Logarithm

log_xy + log_yx² = 3. Find log_xy³.
1. 4
2. 3
3. 3^1/2
4. 3^1/16
Choice B
3
CAT Exponents and Logarithms: Basic identities of Logarithm

If log₂ 4 * log₄ 8 * log₈ 16 * ……………n^th term = 49, what is the value of n?
1. 49
2. 48
3. 34
4. 24
Choice B
48
CAT Exponents and Logarithms: Basic identities of Logarithm

If 3^{3 + 6 + 9 + ……… 3x} = (0.\\overline{037}\\))^-66, what is the value of x?
1. 3
2. 6
3. 7
4. 11
Choice D
11
CAT Exponents and Logarithms: Basic identities of Logarithm

x, y, z are 3 integers in a geometric sequence such that y - x is a perfect cube.
Given, log₃₆x² + log₆√y + 3log₂₁₆y^1/2z = 6. Find the value of x + y + z.
1. 189
2. 190
3. 199
4. 201
Choice A
189
CAT Exponents and Logarithms: Basic identities of Logarithm

10^{log(3 - 10^logy)} = log₂(9 - 2^y), Solve for y.
1. 0
2. 3
3. 0 and 3
4. none of these
Choice D
none of these
CAT Exponents and Logarithms: Value of x

4^{6+12+18+24+…+6x} = (0.0625)^-84, what is the value of x?
1. 7
2. 6
3. 9
4. 12
Choice A
7

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

CAT 2023 Slot 3 - QA

For a real number \(x\), if \(\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}\), and \(\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}\) are in an arithmetic progression, then the common difference is
1. \(\log _4 7\)
2. \(\log _4\left(\frac{3}{2}\right)\)
3. \(\log _4\left(\frac{7}{2}\right)\)
4. \(\log _4\left(\frac{23}{2}\right)\)
Choice C
\(\log _4\left(\frac{7}{2}\right)\)
CAT 2023 Slot 3 - QA

Let \(n\) and \(m\) be two positive integers such that there are exactly 41 integers greater than \(8^m\) and less than \(8^n\), which can be expressed as powers of 2 . Then, the smallest possible value of \(n+m\) is
1. 14
2. 42
3. 16
4. 44
Choice C
16
CAT 2023 Slot 3 - QA

Let \(n\) be any natural number such that \(5^{n-1} \lt 3^{n+1}\). Then, the least integer value of \(m\) that satisfies \(3^{n+1} \lt 2^{n+m}\) for each such \(n\), is
5
CAT 2023 Slot 2 - QA

Let \(a, b, m\) and \(n\) be natural numbers such that \(a>1\) and \(b>1\). If \(a^m b^n=144^{145}\), then the largest possible value of \(n-m\) is
1. 580
2. 290
3. 579
4. 289
Choice C
579
CAT 2023 Slot 2 - QA

For some positive real number \(x\), if \(\log _{\sqrt{3}}(x)+\frac{\log _x(25)}{\log _x(0.008)}=\frac{16}{3}\), then the value of \(\log _3\left(3 x^2\right)\) is
7
CAT 2023 Slot 1 - QA

If \(x\) and \(y\) are positive real numbers such that \(\log _x\left(x^2+12\right)=4\) and \(3 \log _y x=1\), then \(x+y\) equals
1. 20
2. 11
3. 68
4. 10
Choice D
10
CAT 2022 Slot 3 - QA

If \(\left(\sqrt{\frac{7}{5}}\right)^{3 x-y}=\frac{875}{2401}\) and \(\left(\frac{4 a}{b}\right)^{6 x-y}=\left(\frac{2 a}{b}\right)^{y-6 x}\), for all non-zero real values of \(a\) and \(b\), then the value of \(x+y\) is
14
CAT 2022 Slot 2 - QA

The number of distinct integer values of \(n\) satisfying \(\frac{4-\log _2 n}{3-\log _4 n}\lt0\), is
47
CAT 2021 Slot 3 - QA

For a real number a, if \(\frac{\log _{15} a+\log _{32} a}{\left(\log _{15} a\right)\left(\log _{32} a\right)}=\) = 4 then a must lie in the range
1. 4 < a < 5
2. 3 < a < 4
3. a > 5
4. 2 < a < 3
Choice A
4 < a < 5

Correct: 22.74%
Incorrect: 11.08%
Unattempted: 66.18%
CAT 2021 Slot 2 - QA

If log₂[3 + log₃{4 + log₄(x - 1)}] - 2 = 0 then 4x equals
5

Correct: 45.74%
Incorrect: 21.21%
Unattempted: 33.05%
CAT 2021 Slot 1 - QA

If \(5-\log _{10} \sqrt{1+x}+4 \log _{10} \sqrt{1-x}=\log _{10} \frac{1}{\sqrt{1-x^{2}}}\) , then 100 x equals
99

Correct: 11.33%
Incorrect: 29.38%
Unattempted: 59.29%
CAT 2020 Question Paper Slot 3 - Exponents & Powers

If x₁ = -1 and x_m = x_{m + 1} + (m + 1) for every positive integer m, then x₁₀₀ equals
1. -5050
2. -5051
3. -5150
4. -5151
CAT 2020 Question Paper Slot 3 - Logarithms

Let log_a30 = A, log_a\\frac{5}{3}) = -B and log₂a = \\frac{1}{3}), then log₃a equals
1. \\frac{2}{A+B-3})
2. \\frac{A+B-3}{2})
3. \\frac{A+B}{2}) - 3
4. \\frac{2}{A+B}) - 3
CAT 2020 Question Paper Slot 3 - Logarithms

\\frac{2×4×8×16}{(log_{2}4)^{2}(log_{4}8)^{3}(log_{8}16)^{4}}) equals

Correct Answer: 24
CAT 2020 Question Paper Slot 3 - Exponents & Powers

If a,b,c are non-zero and 14^a = 36^b = 84^c, then 6b(\\frac{1}{c}) - \\frac{1}{a})) is equal to

Correct Answer: 3
CAT 2020 Question Paper Slot 2 - Logarithms

The value of log_a\\frac{a}{b}) + log_b\\frac{b}{a}), for 1 < a ≤ b cannot be equal to
1. -0.5
2. 1
3. 0
4. -1
CAT 2020 Question Paper Slot 1 - Logarithms

If log₄ 5 = (log₄ y) (log₆ √5), then y equals

Correct Answer: 36
CAT 2020 Question Paper Slot 1 - Exponents & Powers

The number of real-valued solutions of the equation 2^x + 2^-x = 2 - (x - 2)² is
1. infinite
2. 0
3. 1
4. 2
CAT 2020 Question Paper Slot 1 - Exponents & Powers

If x = (4096)^7+4√3, then which of the following equals 64?
1. \\frac{x^{7/2}}{x^{4/√3}})
2. \\frac{x^{7}}{x^{4√3}})
3. \\frac{x^{7/2}}{x^{2√3}})
4. \\frac{x^{7}}{x^{2√3}})
CAT 2020 Question Paper Slot 1 - Exponents & Powers

If y is a negative number such that 2^y²log₃5 = 5^log₂3, then y equals
1. log₂(1/3)
2. log₂(1/5)
3. −log₂(1/3)
4. −log₂(1/5)
CAT 2019 Question Paper Slot 2 - Logarithms

The real root of the equation 2^6x + 2^3x+2 - 21 = 0 is
1. \\frac{log_{2}3}{3})
2. log₂9
3. \\frac{log_{2}7}{3})
4. log₂27
Choice A
\\frac{log_{2}3}{3})
CAT 2019 Question Paper Slot 2 - Logarithms

If x is a real number ,then \\sqrt{log_{e}\frac{4x - x^2}{3}}) is a real number if and only if
1. -3 ≤ x ≤ 3
2. 1 ≤ x ≤ 2
3. 1 ≤ x ≤ 3
4. -1 ≤ x ≤ 3
Choice C
1 ≤ x ≤ 3
CAT 2019 Question Paper Slot 2 - Exponents & Powers

If 5^x - 3^y = 13438 and 5^x-1 + 3^y+1 = 9686 , then x+y equals [TITA]
13
CAT 2019 Question Paper Slot 1 - Exponents

If (5.55)^x = (0.555)^y = 1000, then the value of \\frac{1}{x}) - \\frac{1}{y}) is
1. 1
2. \\frac{1}{3})
3. \\frac{2}{3})
4. 3
Choice B
\\frac{1}{3})
CAT 2019 Question Paper Slot 1 - Exponents

If m and n are integers such that (\\sqrt{2}))¹⁹ 3⁴ 4² 9^m 8ⁿ = 3ⁿ 16^m (∜64) then m is
1. -16
2. -24
3. -12
4. -20
Choice C
-12
CAT 2019 Question Paper Slot 1 - Logarithms

Let x and y be positive real numbers such that log₅(x + y) + log₅(x - y) = 3, and log₂y - log₂x = 1 - log₂3. Then xy equals
1. 25
2. 150
3. 250
4. 100
Choice B
150
CAT 2018 Question Paper Slot 2 - Logarithm

If p³ = q⁴ = r⁵ = s⁶, then the value of log_s (pqr) is equal to
1. \\frac{24}{5}\\)
2. 1
3. \\frac{47}{10}\\)
4. \\frac{16}{5}\\)
Choice C
\\frac{47}{10}\\)
CAT 2018 Question Paper Slot 2 - Logarithm

\\frac{1}{log_{2}100}\\) - \\frac{1}{log_{4}100}\\) + \\frac{1}{log_{5}100}\\) - \\frac{1}{log_{10}100}\\) + \\frac{1}{log_{20}100}\\) - \\frac{1}{log_{25}100}\\) + \\frac{1}{log_{50}100}\\) = ?
1. 0
2. \\frac{1}{2}\\)
3. -4
4. 10
Choice B
\\frac{1}{2}\\)
CAT 2018 Question Paper Slot 1 - Logarithm

If x is a positive quantity such that 2^x = 3^log₅2 , then x is equal to
1. log₅9
2. 1 + log₅\\frac{3}{5})
3. 1 + log₃\\frac{5}{3})
4. log₅8
Choice B
1 + log₅\\frac{3}{5})
CAT 2018 Question Paper Slot 1 - Logarithm

If log₁₂81 = p, then 3(\\frac{4 - p}{4 + p})) is equal to:
1. log₂8
2. log₆8
3. log₄16
4. log₆16
Choice B
log₆8
CAT 2018 Question Paper Slot 1 - Exponents

Given that x²⁰¹⁸ y²⁰¹⁷ = 1/2 and x²⁰¹⁶ y²⁰¹⁹ = 8, the value of x² + y³ is
1. \\frac{37}{4})
2. \\frac{31}{4})
3. \\frac{35}{4})
4. \\frac{33}{4})
Choice D
\\frac{33}{4})
CAT 2018 Question Paper Slot 1 - Logarithms

If log₂(5 + log₃a) = 3 and log₅(4a + 12 + log₂b) = 3, then a + b is equal to
1. 32
2. 59
3. 67
4. 40
Choice B
59
CAT 2017 Question Paper Slot 2 - Exponents & Logarithms

If x is a real number such that log₃5 = log₅(2 + x), then which of the following is true?
1. 0 < x < 3
2. 23 < x < 30
3. x > 30
4. 3 < x < 23
Choice D
3 < x < 23
CAT 2017 Question Paper Slot 2 - Exponents & Logarithms

If 9^{x - (\\frac{1}{2}))} – 2^{2x – 2} = 4^x – 3^{2x – 3}, then x is
1. \\frac{3}{2}\\)
2. \\frac{2}{5}\\)
3. \\frac{3}{4}\\)
4. \\frac{4}{9}\\)
Choice A
\\frac{3}{2}\\)
CAT 2017 Question Paper Slot 2 - Exponents & Logarithms

If log(2^a × 3^b × 5^c) is the arithmetic mean of log(2² × 3³ × 5), log(2⁶ × 3 × 5⁷), and log(2 × 3² × 5⁴), then a equals [TITA]
3
CAT 2017 Question Paper Slot 1 - Exponents & Logarithms

Suppose, log₃x = log₁₂y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log₆G is equal to:
1. √a
2. 2a
3. \\frac{a}{2})
4. a
Choice D
a
CAT 2017 Question Paper Slot 1 - Exponents & Logarithms

The value of log_0.008√5 + log_√381 – 7 is equal to:
1. \\frac{1}{3})
2. \\frac{2}{3})
3. \\frac{5}{6})
4. \\frac{7}{6})
Choice C
\\frac{5}{6})
CAT 2017 Question Paper Slot 1 - Exponents & Logarithms

If 9^{2x – 1} – 81^x-1 = 1944, then x is
1. 3
2. \\frac{9}{4})
3. \\frac{4}{9})
4. \\frac{1}{3})
Choice B
\\frac{9}{4})

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

XAT 2020 Question Paper - QADI
What is the remainder if 19²⁰ – 20¹⁹ is divided by 7?
1. 5
2. 1
3. 6
4. 0
5. 3
Choice A
5
XAT 2019 Question Paper - QADI
If \\sqrt[3]{7^{a} \times(35)^{b+1} \times(20)^{c+2}}) is a whole number then which one of the statements below is consistent with it?
1. a = 2, b = 1, c = 1
2. a = 1, b = 2, c = 2
3. a = 2, b = 1, c = 2
4. a = 3, b = 1, c = 1
5. a = 3, b = 2, c = 1
Choice E
a = 3, b = 2, c = 1
XAT 2019 Question Paper - QADI
\\frac{\log (97-56 \sqrt{3})}{\log \sqrt{7+4 \sqrt{3}}}) equals which of the following?
1. None of the others
2. -2
3. -4
4. -3
5. -8
Choice C
-4
XAT 2019 Question Paper - QADI
If \x^{2}+x+1=0, \text { then } x^{2018}+x^{2019}) then equals which of the following:
1. x + 1
2. x
3. -x
4. None of the others
5. x - 1
Choice C
-x

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

IPMAT 2020 Sample Paper - IPM Rohtak Quants
Given A = 2⁶⁵ and B = (2⁶⁴ + 2⁶³ + 2⁶² + ... + 2⁰), which of the following is true?
1. B is 2⁶⁴ larger than A
2. A and B are equal
3. B is larger than A by 1
4. A is larger than B by 1
Choice D
A is larger than B by 1
IPMAT 2020 Sample Paper - IPM Rohtak Quants
If log 2, log (2x - 1) and log (2x + 3) are in A.P, then x is equal to ____
1. \\frac{5}{2}\\)
2. log₂5
3. log₃2
4. 32
Choice A
\\frac{5}{2}\\)
IPMAT 2020 Question Paper - IPM Indore Quants
The value of 0.04^{log_√5(\\frac{1}{4}) + \\frac{1}{8}) + \\frac{1}{16}))} is __________.
\\frac{16}{7})
IPMAT 2020 Question Paper - IPM Indore Quants
If log₅log₈(x² - 1) = 0, then a possible value of x is
1. 2√2
2. √2
3. 2
4. 3
Choice D3
IPMAT 2019 Question Paper - IPM Indore Quants
Suppose that a, b, and c are real numbers greater than 1. Then the value of \\frac{1}{1+\log _{a^{2} b} \frac{c}{a}}+\frac{1}{1+\log _{b^{2} c} \frac{a}{b}}+\frac{1}{1+\log _{c^{2} a} \frac{b}{c}}\\) is
3
IPMAT 2019 Question Paper - IPM Indore Quants
If x, y, z are positive real numbers such that x¹² = y¹⁶ = z²⁴,and the three quantities 3log_yx, 4log_zy, nlog_xz are in arithmetic progression, then the value of n is
16
IPMAT 2019 Question Paper - IPM Indore Quants
The inequality \\log _{2} \frac{3x - 1}{2 - x} < 1\\) holds true for
1. x ∈ (\\frac{1}{3}\\), 1)
2. x ∈ (\\frac{1}{3}\\), 2)
3. x ∈ (0, \\frac{1}{3}\\)) ∪ (1,2)
4. x ∈ (-∞, 1)
Choice A
x ∈ (\\frac{1}{3}\\), 1)
IPMAT 2019 Question Paper - IPM Indore Quants
The set of values of x which satisfy the inequality 0.7^{2x² - 3x + 4} < 0.343 is
1. (\\frac{1}{2}\\), 1)
2. (\\frac{1}{2}\\), ∞)
3. (-∞, \\frac{1}{2}\\))
4. (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)
Choice D
(-∞, \\frac{1}{2}\\)) ∪ (1, ∞)
IPMAT 2019 Question Paper - IPM Indore Quants
The value of \\log _{3} 30^{-1} + \log _{4} 900^{-1} + \log _{5} 30^{-1}\\) is
1. 0.5
2. 30
3. 2
4. 1
Choice D
1
IPMAT 2019 Question Paper - IPM Indore Quants
The inequality \\log _{a}{f(x)} < \log _{a}{g(x)}\\) implies that
1. f(x) > g(x) > 0 for 0 < a < 1 and g(x) > f(x) > 0 for a > 1
2. g(x) > f(x) > 0 for 0 < a < 1 and f(x) > g(x) > 0 for a > 1
3. f(x) > g(x) > 0 for a > 0
4. g(x) > f(x) > 0 for a > 0
Choice A
\(f(x)>g(x)>0\) for \(0f(x)>0\) for \(a>1\)
IPMAT 2019 Question Paper - IPM Indore Quants
Determine the greatest number among the following four numbers
1. 2³⁰⁰
2. 3²⁰⁰
3. 2¹⁰⁰ + 3¹⁰⁰
4. 4¹⁰⁰
Choice B
3²⁰⁰