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Algebra • Logarithms• Official Answer Keys

CAT Logarithms & Exponential Equations Questions

30+ Official CAT Log property questions, base-change formulas, and inequality solving. Official questions curated from CAT 2017–2025 with detailed explanations.

32 Total Questions
MCQ: 22 (+3 / -1)
TITA: 10 (0 Negative Penalty)
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Official Exam Questions & Explanations (25 of 32)

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Question 1 of 32
CAT 2025 Slot 2QAModern Math • LogarithmsHard
If log⁡64x2+log⁡8y+3log⁡512(yz)=4\log_{64} x^2 + \log_8 \sqrt{y} + 3\log_{512}\left(\sqrt{yz}\right) = 4, where x,yx, y and zz are positive real numbers, then the minimum possible value of (x+y+z)(x+y+z) is
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Question 2 of 32
CAT 2025 Slot 1QAModern Math • LogarithmsHard
The number of distinct integers nn for which log⁡1/4(n2−7n+11)>0\log_{1/4}(n^2 - 7n + 11) > 0, is
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Question 3 of 32
CAT 2024 Slot 1QAModern Math • LogarithmsModerate
If xx is a positive real number such that 4log⁡10x+4log⁡100x+8log⁡1000x=134 \log _{10} x+4 \log _{100} x+8 \log _{1000} x=13, t hen greatest integer not exceeding x , is
TITA Answer:
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Question 4 of 32
CAT 2024 Slot 2QAModern Math • LogarithmsHard
If a, b and c are positive real numbers such that a>10≥b≥ca > 10 \ge b \ge c and log⁡8(a+b)log⁡2c+log⁡27(a−b)log⁡3c=23\frac{\log_8(a+b)}{\log_2 c} + \frac{\log_{27}(a - b)}{\log_3 c} = \frac{2}{3}, then the greatest possible integer value of a is
TITA Answer:
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Question 5 of 32
CAT 2024 Slot 3QAModern Math • LogarithmsHard
The sum of all distinct real values of xx that satisfy the equation 10x+410x=81210^x + \frac{4}{10^x} = \frac{81}{2}, is
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Question 6 of 32
CAT 2023 Slot 1QAModern Math • LogarithmsModerate
If xx and yy are positive real numbers such that log⁡x(x2+12)=4\log_{x}\left(x^{2}+12\right)=4 and 3log⁡yx=13 \log _{y} x=1, then x+yx+y equals
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Question 7 of 32
CAT 2025 Slot 3QAModern Math • LogarithmsHard
The sum of all possible real values of xx for which log⁡x−3(x2−9)=log⁡x−3(x+1)+2\log_{x-3}(x^2-9)=\log_{x-3}(x+1)+2, is
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Question 8 of 32
CAT 2023 Slot 2QAModern Math • LogarithmsModerate
For some positive real number xx, if log⁡3(x)+log⁡x(25)log⁡x(0.008)=163\log_{\sqrt{3}}(x) + \frac{\log_x (25)}{\log_x (0.008)} = \frac{16}{3}, then the value of log⁡3(3x2)\log_3(3x^2) is
TITA Answer:
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Question 9 of 32
CAT 2023 Slot 3QAModern Math • LogarithmsHard
For a real number xx, if 12,log⁡3(2x−9)log⁡34\frac{1}{2}, \frac{\log _{3}\left(2^{x}-9\right)}{\log _{3} 4}, and log⁡5(2x+172)log⁡54\frac{\log _{5}\left(2^{x}+\frac{17}{2}\right)}{\log _{5} 4} are in an arithmetic progression, then the common difference is
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Question 10 of 32
CAT 2022 Slot 2QAModern Math • LogarithmsModerate
The number of distinct integer values of n satisfying 4−log⁡2n3−log⁡4n<0\frac{4-\log_2 n}{3-\log_4 n} < 0, is
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Question 11 of 32
CAT 2021 Slot 1QAModern Math • LogarithmsModerate
If 5−log⁡101+x+4log⁡101−x=log⁡1011−x25-\log _{10} \sqrt{1+x}+4 \log _{10} \sqrt{1-x}=\log _{10} \frac{1}{\sqrt{1-x^{2}}},then 100x100 \mathrm{x} equals
TITA Answer:
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Question 12 of 32
CAT 2021 Slot 2QAModern Math • LogarithmsHard
For all possible integers nn satisfying 2.25≤2+2n+2≤2022.25 \leq 2+2^{n+2} \leq 202, the number of integer values of 3+3n+13+3^{n+1} is
TITA Answer:
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Question 13 of 32
CAT 2021 Slot 2QAModern Math • LogarithmsModerate
If log⁡2[3+log⁡3{4+log⁡4(x−1)}]−2=0\log_2 [3 + \log_3 \{4 + \log_4 (x - 1)\}] - 2 = 0, then 4x4x equals
TITA Answer:
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Question 14 of 32
CAT 2021 Slot 3QAModern Math • LogarithmsModerate
For a real number aa, if log⁡15a+log⁡32a(log⁡15a)(log⁡32a)=4\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.
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Question 15 of 32
CAT 2020 Slot 1QAModern Math • LogarithmsModerate
If y is a negative number such that 2y2log35=5log⁡232^{y^2log_3 5} = 5^{\log_2 3}, then y equals
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Question 16 of 32
CAT 2020 Slot 1QAModern Math • LogarithmsModerate
If log⁡45=(log⁡4y)(log⁡65)\log _{4} 5=\left(\log _{4} y\right)\left(\log _{6} \sqrt{ } 5\right), then yy equals
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Question 17 of 32
CAT 2020 Slot 2QAModern Math • LogarithmsModerate
The value of log⁡a(ab)+log⁡b(ba)\log_a \left(\frac{a}{b}\right) + \log_b \left(\frac{b}{a}\right), for 1<a≤b1 < a \leq b cannot be equal to
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Question 18 of 32
CAT 2020 Slot 3QAModern Math • LogarithmsHard
If log⁡a30=A\log_a 30 = A, log⁡a(5/3)=−B\log_a(5/3) = - B and log⁡2a=1/3\log_2 a = 1 /3, then log⁡3a\log_3 a equals
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Question 19 of 32
CAT 2020 Slot 3QAModern Math • LogarithmsModerate
$2×4×8×16(log⁡24)2(log⁡48)3(log⁡816)4\small \dfrac{2\times4\times8\times16}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4} is equal to
TITA Answer:
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Question 20 of 32
CAT 2019 Slot 2QAModern Math • LogarithmsModerate
The real root of the equation 26x+23x+2−21=02^{6x} + 2^{3x+2} - 21 = 0 is
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Question 21 of 32
CAT 2019 Slot 2QAModern Math • LogarithmsHard
If xx is a real number, then log⁡e4x−x23\sqrt{\log_e \frac{4x-x^2}{3}} is a real number if and only if
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Question 22 of 32
CAT 2018 Slot 1QAModern Math • LogarithmsModerate
If xx is a positive quantity such that 2x=3log⁡522^x = 3^{\log_5 2}, then xx is equal to
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Question 23 of 32
CAT 2018 Slot 1QAModern Math • LogarithmsHard
If log⁡1281=p\log_{12}81 = p, then 34−p4+p3\frac{4-p}{4+p} is equal to
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Question 24 of 32
CAT 2018 Slot 1QAModern Math • LogarithmsModerate
If log⁡2(5+log⁡3a)=3\log _{2}\left(5+\log _{3} a\right)=3 and log⁡5(4a+12+log⁡2b)=3\log _{5}\left(4 a+12+\log _{2} b\right)=3, then a+ba+b is equal to
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Question 25 of 32
CAT 2018 Slot 2QAModern Math • LogarithmsModerate
The smallest integer nn for which 4n>17194^n > 17^{19} holds, is closest to
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Frequently Asked Questions about CAT Logarithms & Exponential Equations Questions

Why are Logarithms questions crucial for scoring 99+ percentile in CAT?

CAT convenors place heavy emphasis on testing core intuition in Logarithms problems. Mastering standard patterns allows candidates to secure +3 marks with high accuracy and minimal time.

Are these questions from real CAT exam slots?

Yes, 100% of the questions in this module are official previous year questions administered in CAT 2017 through CAT 2025.

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