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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.

  1. CAT Exponents and Logarithms: Inequalities

    If log2X + log4X = log0.25√6 and x > 0, then x is

    1. 6-1/6
    2. 61/6
    3. 3-1/3
    4. 61/3
    Choice A
    6-1/6

  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
    Choice D
    2

  3. CAT Exponents and Logarithms: Quadratic Equations

    If 22x+4 – 17 × 2x+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

  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}\\)
    Choice D
    \\frac{2(b + 3)}{3b}\\)

  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
    Choice A
    214

  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 logx10?

    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}\\)

  7. CAT Exponents and Logarithms: Basic identities of Logarithm

    log3x + logx3 = \\frac{17}{4}\\). Find the value of x.

    1. 34
    2. 31/4
    3. 34 or 31/4
    4. 31/3
    Choice C
    34 or 31/4

  8. CAT Exponents and Logarithms: Basic identities of Logarithm

    logxy + logyx2 = 3. Find logxy3.

    1. 4
    2. 3
    3. 31/2
    4. 31/16
    Choice B
    3

  9. CAT Exponents and Logarithms: Basic identities of Logarithm

    If log2 4 * log4 8 * log8 16 * ……………nth term = 49, what is the value of n?

    1. 49
    2. 48
    3. 34
    4. 24
    Choice B
    48

  10. CAT Exponents and Logarithms: Basic identities of Logarithm

    If 33 + 6 + 9 + ……… 3x = (0.\\overline{037}\\))-66, what is the value of x?

    1. 3
    2. 6
    3. 7
    4. 11
    Choice D
    11

  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 + 3log216y1/2z = 6. Find the value of x + y + z.

    1. 189
    2. 190
    3. 199
    4. 201
    Choice A
    189

  12. CAT Exponents and Logarithms: Basic identities of Logarithm

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

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

  13. CAT Exponents and Logarithms: Value of x

    46+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.


  1. 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%

  2. 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%

    1. 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%

      1. CAT 2020 Question Paper Slot 3 - Exponents & Powers

        If x1 = -1 and xm = xm + 1 + (m + 1) for every positive integer m, then x100 equals

        1. -5050
        2. -5051
        3. -5150
        4. -5151
        Choice A
        -5050

      2. CAT 2020 Question Paper Slot 3 - Logarithms

        Let loga30 = A, loga\\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
        Choice A
        \\frac{2}{A+B-3})

      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

        24

      4. CAT 2020 Question Paper Slot 3 - Exponents & Powers

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

        3

      5. CAT 2020 Question Paper Slot 2 - Logarithms

        The value of loga\\frac{a}{b}) + logb\\frac{b}{a}), for 1 < a ≤ b cannot be equal to

        1. -0.5
        2. 1
        3. 0
        4. -1
        Choice B
        1

      6. CAT 2020 Question Paper Slot 1 - Logarithms

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

        36

      7. CAT 2020 Question Paper Slot 1 - Exponents & Powers

        The number of real-valued solutions of the equation 2x + 2-x = 2 - (x - 2)2 is

        1. infinite
        2. 0
        3. 1
        4. 2
        Choice B
        0

      8. 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}})
        Choice C
        \\frac{x^{7/2}}{x^{2√3}})

      9. CAT 2020 Question Paper Slot 1 - Exponents & Powers

        If y is a negative number such that 2y2log35 = 5log23, then y equals

        1. log(1/3)
        2. log(1/5)
        3. −log(1/3)
        4. −log(1/5)
        Choice A
        log(1/3)

      10. CAT 2019 Question Paper Slot 2 - Logarithms

        The real root of the equation 26x + 23x+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})

      11. 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

      12. CAT 2019 Question Paper Slot 2 - Exponents & Powers

        If 5x – 3y = 13438 and 5x-1 + 3y+1 = 9686 , then x+y equals [TITA]

          13

        1. 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})

        2. CAT 2019 Question Paper Slot 1 - Exponents

          If m and n are integers such that (\\sqrt{2}))19 34 42 9m 8n = 3n 16m (∜64) then m is

          1. -16
          2. -24
          3. -12
          4. -20
          Choice C
          -12

        3. 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
          -->

        4. CAT 2018 Question Paper Slot 2 - Logarithm

          If p3 = q4 = r5 = s6, then the value of logs (pqr) is equal to

          1. \\frac{24}{5}\\)
          2. 1
          3. \\frac{47}{10}\\)
          4. \\frac{16}{5}\\)
          Choice C
          \\frac{47}{10}\\)

        5. 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}\\)

        6. CAT 2018 Question Paper Slot 1 - Logarithm

          If x is a positive quantity such that 2x = 3log52 , 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})

        7. 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

        8. 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})

        9. 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

        10. 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

        11. CAT 2017 Question Paper Slot 2 - Exponents & Logarithms

          If 9x - (\\frac{1}{2})) – 22x – 2 = 4x – 32x – 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}\\)

        12. CAT 2017 Question Paper Slot 2 - Exponents & Logarithms

          If log(2a × 3b × 5c) is the arithmetic mean of log(22 × 33 × 5), log(26 × 3 × 57), and log(2 × 32 × 54), then a equals [TITA]

            3

          1. 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

          2. CAT 2017 Question Paper Slot 1 - Exponents & Logarithms

            The value of log0.008√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})

          3. CAT 2017 Question Paper Slot 1 - Exponents & Logarithms

            If 92x – 1 – 81x-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.


          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
            Choice A
            5

          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
            Choice E
            a = 3, b = 2, c = 1

          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
            Choice C
            -4

          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.


          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
            Choice D
            A is larger than B by 1

          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
            Choice A
            \\frac{5}{2}\\)

          3. IPMAT 2020 Question Paper - IPM Indore Quants

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

            \\frac{16}{7})

          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
            Choice D3

          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


          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 3logyx, 4logzy, nlogxz are in arithmetic progression, then the value of n is


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

          8. IPMAT 2019 Question Paper - IPM Indore Quants

            The set of values of x which satisfy the inequality 0.72x2 - 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, ∞)

          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
            Choice D
            1

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

          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
            Choice B
            3200


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