CAT Questions | CAT Arithmetic

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.

1. 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

   Correct AnswerChoice A
   6^-1/6

   Explanation

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2. 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

   Correct AnswerChoice D
   2

   Explanation Video Solution

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3. 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

   Correct AnswerChoice C
   x can be either a positive value or a negative value

   Explanation Video Solution

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4. 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}\\)

   Correct AnswerChoice D
   \\frac{2(b + 3)}{3b}\\)

   Explanation Video Solution

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5. 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

   Correct AnswerChoice A
   214

   Explanation Video Solution

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6. 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}\\)

   Correct AnswerChoice A
   \\frac{a + b}{2ab}\\)

   Explanation

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7. 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

   Correct AnswerChoice C
   3⁴ or 3^1/4

   Explanation

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8. 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

   Correct AnswerChoice B
   3

   Explanation

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9. 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

   Correct AnswerChoice B
   48

   Explanation

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10. 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

    Correct AnswerChoice D
    11

    Explanation

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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

    Correct AnswerChoice A
    189

    Explanation

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12. CAT Exponents and Logarithms: Basic identities of Logarithm

    10^{log(3 - 10logy)} = log₂(9 - 2^y), Solve for y.

    1. 0
    2. 3
    3. 0 and 3
    4. none of these

    Correct AnswerChoice D
    none of these

    Explanation

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13. 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

    Correct AnswerChoice A
    7

    Explanation

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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.

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1. 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

   Correct Answer14

   Video ExplanationExplanation

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2. 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

   Correct Answer47

   Video ExplanationExplanation

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3. 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

   Correct AnswerChoice A
   4 < a < 5

   Video ExplanationExplanation

   Check StatsCorrect: 22.74%
   Incorrect: 11.08%
   Unattempted: 66.18%

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4. CAT 2021 Slot 2 - QA

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

   Correct Answer5

   Video ExplanationExplanation

   Check StatsCorrect: 45.74%
   Incorrect: 21.21%
   Unattempted: 33.05%

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5. 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

   Correct Answer99

   Video ExplanationExplanation

   Check StatsCorrect: 11.33%
   Incorrect: 29.38%
   Unattempted: 59.29%

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6. 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

   Correct AnswerChoice A
   -5050

   Video SolutionExplanation

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7. 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

   Correct AnswerChoice A
   \\frac{2}{A+B-3})

   Video SolutionExplanation

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8. 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 Answer24
   

   Video SolutionExplanation

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9. 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 Answer3
   

   Video SolutionExplanation

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10. 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

    Correct AnswerChoice B
    1

    Video SolutionExplanation

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11. CAT 2020 Question Paper Slot 1 - Logarithms

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

    Correct Answer36
    

    Video SolutionExplanation

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12. 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

    Correct AnswerChoice B
    0

    Video SolutionExplanation

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13. 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}})

    Correct AnswerChoice C
    \\frac{x^{7/2}}{x^{2√3}})

    Video SolutionExplanation

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14. 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)

    Correct AnswerChoice A
    log₂ (1/3)

    Video SolutionExplanation

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15. 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

    Correct AnswerChoice A
    \\frac{log_{2}3}{3})

    ExplanationVideo Solution

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16. 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

    Correct AnswerChoice C
    1 ≤ x ≤ 3

    ExplanationVideo Solution

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17. 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]

    Correct Answer13
    

    ExplanationVideo Solution

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18. 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

    Correct AnswerChoice B
    \\frac{1}{3})

    ExplanationVideo Solution

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19. 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

    Correct AnswerChoice C
    -12

    ExplanationVideo Solution

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20. 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

    Correct AnswerChoice B
    150

    ExplanationVideo Solution -->

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21. 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}\\)

    Correct AnswerChoice C
    \\frac{47}{10}\\)

    ExplanationVideo Solution

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22. 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

    Correct AnswerChoice B
    \\frac{1}{2}\\)

    ExplanationVideo Solution

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23. 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

    Correct AnswerChoice B
    1 + log₅\\frac{3}{5})

    ExplanationVideo Solution

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24. 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

    Correct AnswerChoice B
    log₆8

    ExplanationVideo Solution

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25. 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})

    Correct AnswerChoice D
    \\frac{33}{4})

    ExplanationVideo Solution

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26. 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

    Correct AnswerChoice B
    59

    ExplanationVideo Solution

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27. 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

    Correct AnswerChoice D
    3 < x < 23

    ExplanationVideo Solution

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28. 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}\\)

    Correct AnswerChoice A
    \\frac{3}{2}\\)

    ExplanationVideo Solution

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29. 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]

    Correct Answer3
    

    ExplanationVideo Solution

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30. 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

    Correct AnswerChoice D
    a

    ExplanationVideo Solution

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31. 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})

    Correct AnswerChoice C
    \\frac{5}{6})

    ExplanationVideo Solution

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32. 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})

    Correct AnswerChoice B
    \\frac{9}{4})

    ExplanationVideo Solution

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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.

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1. 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

   Correct AnswerChoice A
   5

   Video SolutionExplanation

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2. 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

   Correct AnswerChoice E
   a = 3, b = 2, c = 1

   Video SolutionExplanation

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3. 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

   Correct AnswerChoice C
   -4

   Video SolutionExplanation

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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

   Correct AnswerChoice C
   -x

   Video SolutionExplanation

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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.

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1. 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

   Correct AnswerChoice D
   A is larger than B by 1

   ExplanationVideo Solution

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2. 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

   Correct AnswerChoice A
   \\frac{5}{2}\\)

   Explanation

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3. IPMAT 2020 Question Paper - IPM Indore Quants

   The value of 0.04^{log_√5(\\frac{1}{4}) + \\frac{1}{8}) + \\frac{1}{16}))} is __________.

   Correct Answer\\frac{16}{7})
   

   Explanation

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4. 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

   Correct AnswerChoice D3
   

   Explanation

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5. 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

   Correct Answer3

   ExplanationVideo Solution

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6. 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

   Correct Answer16

   ExplanationVideo Solution

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7. 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)

   Correct AnswerChoice A
   x ∈ (\\frac{1}{3}\\), 1)

   ExplanationVideo Solution

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8. 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, ∞)

   Correct AnswerChoice D
   (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)

   ExplanationVideo Solution

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9. 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

   Correct AnswerChoice D
   1

   ExplanationVideo Solution

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10. 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

    Correct AnswerChoice A
    \(f(x)>g(x)>0\) for \(0f(x)>0\) for \(a>1\)

    ExplanationVideo Solution

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11. 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

    Correct AnswerChoice B
    3²⁰⁰

    ExplanationVideo Solution

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