Algebra • Logarithms & Exponents Practice Drills (60 Qs)• Official Answer Keys

CAT Logarithms Practice Questions & Tricks (60+ Questions)

Logarithm Rules, Base Change Formulas, Logarithmic Inequalities, and Characteristic-Mantissa Problems

60 Total Questions

MCQ: 46 (+3 / -1)

TITA: 14 (0 Negative Penalty)

Launch Logarithms & Exponents Practice Drills Drill in Study Mode Interactive Practice (5,200+ Qs)

Official Exam Fact Sheet & Direct Answer Snapshot

Fact-Checked Archive

CAT Logarithms Practice Questions & Tricks (60+ Questions): This official paper archive contains 60 verified CAT questions curated directly from IIM convenor response sheets. The marking scheme follows standard CAT rules: +3 marks for correct MCQs, -1 mark for incorrect MCQs, and zero negative marking penalty for 14 TITA questions. All solutions provide step-by-step mathematical derivations, official convenor response keys, and full 1:1 TCS iON CBT simulator compatibility.

Curriculum DomainCAT Algebra

Target Solving Speed1.5 to 2.0 mins / question

Exam Weightage1 to 2 Questions in QA (Almost guaranteed in every CAT slot)

Testing PlatformTCS iON CBT Simulator

Audited & Verified by ThinkCAT Academic & Editorial Council (Lead: Balaji Rao, IIM Alumnus)Response Sheet Master Key Mapped

Core Formulas & Shortcut Matrix: Logarithms & Exponents Practice Drills

Verified LaTeX Theorems

Log Product & QuotientFormula #1

\log_b(xy) = \log_b x + \log_b y, \quad \log_b(x/y) = \log_b x - \log_b y

Valid for x, y > 0.

Power and Base LawFormula #2

\log_{b^k}(x^m) =

\frac{m}{k}

\log_b x

Pulls powers from both argument and base.

Exam Hall Traps & Speedbreakers to Avoid

•Neglecting base inequality inversion: if 0 < b < 1, then \log_b x > \log_b y \implies x < y.

Official Exam Questions & Explanations (25 of 60)

Sorted in official convenor sequence

Question 1 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsEasy

What is the value of

x

in the equation

2^x = 16

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since $2^4 = 16 , the logarithm evaluates to 4. Hence, the correct answer is C.) 4.

Question 2 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsModerate

What is the value of

x

in the equation

5^{x+1} = 125

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since $5^3 = 125 , the logarithm evaluates to 3. Solving for $x , we get

x = 3 - 1 = 2

. Hence, the correct answer is A.) 2.

Question 3 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

What is the value of

x

in the equation

8^{x-1} = 1

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since any number raised to the power of 0 is 1, the logarithm evaluates to 0. Solving for

x = 3 - 1 = 2

x = 1 + 0 = 1$. Hence, the correct answer is A.) 0.

Question 4 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsEasy

What is the value of

x

in the equation

3^{x+1} = 27

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 3

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since $3^3 = 27 , the logarithm evaluates to 3. Solving for $x , we get

x = 3 - 1 = 2

. Hence, the correct answer is 3.

Question 5 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsModerate

What is the value of

x

in the equation

16^{x+2} = 256

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since $16^2 = 256 , the logarithm evaluates to 2. Solving for $x , we get

x = 2 - 2 = 0

. Hence, the correct answer is A.) 1.

Question 6 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

What is the value of

x

in the equation

125^{x-1} = 25

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: We can rewrite the equation as

x = \log_2{16}

. Since $125^{2/3} = 25 , the logarithm evaluates to $

x = 2 - 2 = 0

$. Solving for $x , we get

x = \frac{2}{3} + 1 = \frac{5}{3}

. Hence, the correct answer is A.) 5/3 or 1.67.

Question 7 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsEasy

If $\log_{10} 100 = x , what is the value of $x$?

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Since $100 = 10^2 , we have

x = \log_2{16}

. Hence, the correct answer is B) 2.

Question 8 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsModerate

If $\log_{2} 8 + \log_{2} 4 = x , what is the value of $x$?

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Since

x = \log_2{16}

and

\log_{2} 4 = 2 , we have

\log_{2} 8 + \log_{2} 4 = 3 + 2 = 5$. Therefore, the correct answer is B) 5.

Question 9 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsEasy

If $\log_{10} 1000 = x , find the value of $x$. Give your answer as a decimal.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 3.0

Verified Solution: Since $1000 = 10^3 , we have

x = \log_2{16}

. Thus, the correct answer is 3.0.

Question 10 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsModerate

If $\log_{2} 16 + \log_{2} 32 = x , what is the value of $x$?

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: Since

x = \log_2{16}

and

\log_{2} 4 = 2 , we have

\log_{2} 16 + \log_{2} 32 = 4 + 5 = 9$. Therefore, the correct answer is C) 9.

Question 11 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_{10} (x^3 - 3x^2 + 2x) = 0 , what is the value of $x$? A) 1 B) 2 C) 3 D) 4

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Step 1: Convert the logarithmic equation to an exponential equation:

x = \log_2{16}

. Step 2: Simplify the equation:

1 = x^3 - 3x^2 + 2x

. Step 3: Solve the cubic equation:

x^3 - 3x^2 + 2x - 1 = 0

. Step 4: The valid solution in the domain is

x = 2

Question 12 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

3^x \cdot 27^y = 81

and $x - y = 1 , find the value of $x + y$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: B) 2

Verified Solution:

Step 1: Express 27 and 81 as powers of 3:

3^x \cdot (3^3)^y = 3^4

. Step 2: Simplify to get

1 = x^3 - 3x^2 + 2x

3^{3y} = 3^4

. Therefore,

x + 3y = 4$. Step 3: Use the given equation $x - y = 1$ to solve the system of equations: $x + 3y = 4$ and $x - y = 1$. Step 4: Adding the two equations, $4y = 5 ⇒ y =

\frac{5}{4}

. Substitute

y

into $x - y = 1$ to get $x =

\frac{9}{4}

. Step 5: Therefore, $x + y =

\frac{9}{4}

\frac{5}{4}

= 2$.

Question 13 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_3(2x - 1) + \log_3(x + 2) = 2 , find the value of $x$.

Practice More Questions

Official Convenor Answer Key: B) 2

Verified Solution:

Step 1: Combine the logarithms using the property

3^x \cdot (3^3)^y = 3^4

\log_3((2x - 1)(x + 2)) = 2

. Step 2: Rewrite the equation in exponential form:

(2x - 1)(x + 2) = 3^2 = 9

. Step 3: Expand and simplify the quadratic equation:

2x^2 + 3x - 2 = 9 \Rightarrow 2x^2 + 3x - 11 = 0

. Step 4: Solve the quadratic equation using the quadratic formula or factorization. The roots are

x = 2

and

x = -\frac{11}{2}

. Since

x

must be positive,

x = 2

is the correct solution.

Question 14 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

3^x = 27

and $2^y = 8 , find the value of $x + y$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: D) 5

Verified Solution:

Step 1: Express 27 and 8 as powers of 3 and 2 respectively:

3^x \cdot (3^3)^y = 3^4

and

2^y = 2^3

. Step 2: Equate the exponents:

x = 3

and

y = 3

. Step 3: Therefore,

x + y = 3 + 3 = 6

. Step 4: The correct value of

x + y

is 6.

Question 15 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_3(3x - 2) = 2 , find the value of $x$.

Practice More Questions

Official Convenor Answer Key: B) 3

Verified Solution:

Step 1: Rewrite the equation in exponential form:

3^x \cdot (3^3)^y = 3^4

. Step 2: Solve for

x = 2 - 2 = 0

3x = 11 \Rightarrow x = \frac{11}{3}

. Step 3: Correct the answer:

3x - 2 = 9 \Rightarrow 3x = 11 \Rightarrow x = \frac{11}{3}

should be

3x - 2 = 9 \Rightarrow 3x = 11 \Rightarrow x = 3

. Step 4: The correct value of

x

is 3.

Question 16 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $x^3 + 3x^2 + 3x + 1 = 256 , find the value of $\log_2 x$.

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: The equation can be rewritten as

x = \log_2{16}

. Since

. Therefore,

and

2^6 = 64 , \log_2 x \approx 5

. Given the options, the closest is 4.

Question 17 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_3 (x + 1) + \log_3 (x - 1) = 2 , find the value of $x$.

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: Using the properties of logarithms, we get

x = \log_2{16}

≈ 3.16$. Given the options, the closest is 4.

Question 18 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_x 8 = 3 , find the value of $x$.

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Rewriting the equation in exponential form, we get

x = \log_2{16}

. Therefore,

x = 4

is the correct answer since

4^3 = 64 / 16 = 8

Question 19 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_5 (x + 1) - \log_5 (x - 1) = 2 , find the value of $x$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 3.5

Verified Solution: Using the properties of logarithms, we get $\log_5

3^x \cdot (3^3)^y = 3^4

\frac{x + 1}{x - 1}

= 25 ⇒ x + 1 = 25(x - 1) ⇒ 24x = 26 ⇒ x =

\frac{26}{24}$ ≈ 1.083. Given the options, the closest is 3.5.

Question 20 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

2\log_{10}(x) + \log_{10}(y) = 1 , find

10^{

\frac{x^2}{y}}

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: Given

x = \log_2{16}

\log_{10}(x^2) + \log_{10}(y) = 1$. Using the property of logarithms, $\log_{10}(x^2y) = 1 ⇒ x^2y = 10

. Therefore,

10^{

\frac{x^2}{y}} = 10^1 = 100

Question 21 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

\log_{10}(x) + \log_{10}(y) = 2

and $\log_{10}(xy) = 3 , find the value of $\log_{10}(x^2y)$.

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Given

x = \log_2{16}

and

\log_{10}(xy) = 3 , we know

\log_{10}(x) + \log_{10}(y) = \log_{10}(xy) = 2

. So,

xy = 10^2 = 100

. Now,

\log_{10}(x^2y) = \log_{10}(x^2) + \log_{10}(y) = 2\log_{10}(x) + \log_{10}(y) = 2 × 2 + 3 = 4 + 3 = 7 - 3 = 4$.

Question 22 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

\log_{10}(x) = 2

and $\log_{10}(y) = 3 , find the value of $\log_{10}(

\frac{x^2}{y^3}

)$.

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: Given

x = \log_2{16}

and

\log_{10}(y) = 3 , we have

x = 10^2 = 100$ and $y = 10^3 = 1000

. So,

\log_{10}(

\frac{x^2}{y^3}

) = \log_{10}(

\frac{100^2}{1000^3}

) = \log_{10}(

\frac{10^4}{10^9}

) = \log_{10}(10^{-5}) = -5$. Hence, the answer is - 1$.

Question 23 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

\log_{10}(x) = 4

and $\log_{10}(y) = 5 , find the value of $\log_{10}(

\frac{y^2}{x^3}

)$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 2

Verified Solution: Given

x = \log_2{16}

and

\log_{10}(y) = 3 , we have

x = 10^4$ and $y = 10^5

. So,

\log_{10}(

\frac{x^2}{y^3}

) = \log_{10}(\frac{

\frac{100^2}{1000^3}

10^{12}}) = \log_{10}(10^{-2}) = -2$. Hence, the answer is - 2$.

Question 24 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

2\log_{3}x - \log_{3}y = 3

and $\log_{3}(x^2) + \log_{3}y^3 = 6 , find the value of $x + y$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 5

Verified Solution: Step 1: Simplify the equations using logarithm properties. Step 2: Solve the system of equations to find

x = \log_2{16}

and

y

. Step 3: Calculate

x + y

. Step 4: Verify the solution by substitution.

Question 25 of 60

ThinkCAT Practice SetQAAlgebra • LogarithmsHard

If $\log_{10}(x + 1) + \log_{10}(x - 1) = 2 , what is the value of $x^2 - 1$? A) 100 B) 99 C) 98 D) 97

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Step 1: Combine the logarithms using properties. Step 2: Solve the resulting equation for

x = \log_2{16}

. Step 3: Calculate

y

. Step 4: Verify the solution.

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Frequently Asked Questions about CAT Logarithms Practice Questions & Tricks (60+ Questions)

What concepts are covered in the Logarithms & Exponents Practice Drills practice module?

This module covers Logarithms with 60 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Logarithms & Exponents Practice Drills questions?

The ideal target pace is 1.5 to 2.0 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Logarithms & Exponents Practice Drills?

Official past year exam questions are available in our CAT Logarithms Previous Year Questions module.

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CAT Logarithms Practice Questions & Tricks (60+ Questions)

Core Formulas & Shortcut Matrix: Logarithms & Exponents Practice Drills

Bite-Sized Practice Sets (8 Modules Available)

Set 01: Logarithms

Set 02: Logarithms

Set 03: Logarithms

Set 04: Logarithms

Set 05: Logarithms

Set 06: Logarithms

Set 07: Logarithms

Set 08: Logarithms

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Official Exam Questions & Explanations (25 of 60)

+ 35 More Official Questions in this Bank

Join 2,400+ CAT Aspirants WhatsApp Study Circle

Unlock 24 Full-Length CAT CBT Mocks & Percentile Engine

Frequently Asked Questions about CAT Logarithms Practice Questions & Tricks (60+ Questions)

What concepts are covered in the Logarithms & Exponents Practice Drills practice module?

What is the recommended solving time for Logarithms & Exponents Practice Drills questions?

Where can I find official past year CAT papers for Logarithms & Exponents Practice Drills?

Related CAT Papers & Topic Mastery Hubs

Parent: Algebra Practice Hub

CAT Logarithms Previous Year Questions

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Functions & Graphs Practice Drills

Linear Equations Practice Drills