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Exponents, Logarithms and Infinite Series PYQs for GATE DA

Solve 3+ Exponents, Logarithms and Infinite Series previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

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Question 1

2026 PYQ

Level 3: Exam Standard

Consider two distinct positive real numbers m, n, with m > n . Let x = n^{l o g_{10} (m)} and y = m^{l o g_{10} (n)} . The relation between x and y is _______.

Question 2

2025 PYQ

Level 3: Exam Standard

If a real variable x satisfies 3^{x^{2}} = 27 \times 9^{x}, then the value of \frac{2 ^{x^{2}}}{( 2 ^{x} ) ^{2}} is:

Question 3

2024 PYQ

Level 3: Exam Standard

The sum of the following infinite series is 2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots

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Exponents, Logarithms and Infinite Series PYQs for GATE DA

Solve 3+ Exponents, Logarithms and Infinite Series previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Exponents, Logarithms and Infinite Series

Chapter Roadmap

This chapter builds the algebraic foundation required for quantitative aptitude in Data Analytics.

1. Exponent and Logarithm Laws (Current)

Focus: Rules of indices, change of base, and identities.
Goal: Simplify exponential equations and compare magnitudes.

2. Infinite Series Summation

Focus: Geometric and telescoping sums.
Goal: Evaluate limits and sum infinite terms.

What you will master:

Rapid simplification of exponential terms.
Solving equations where variables are in exponents.
Evaluating infinite sums using standard formulas.

The Logic of Exponents

An exponent $a^{n}$ represents repeated multiplication of the base $a$ . It compresses large multiplications into compact notation.

Core Intuition: Operation Conversion

Multiplication → Addition

a^{m} \cdot a^{n} = a^{m + n}

Division → Subtraction

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Powers → Multiplication

(a^{m})^{n} = a^{mn}

Understanding this conversion is essential for simplifying complex algebraic structures in data science algorithms.

Exponents, Logarithms and Infinite Series: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Quantitative Aptitude · 2026 MCQ

Consider two distinct positive real numbers

m, n

, with

m > n

.
Let

x = n^{l o g_{10} (m)}

and

y = m^{l o g_{10} (n)}

. The relation between

x

and

y

is _______.

A.
$x > y$
B.
$x < y$
C.
$x = y$
D.
$x = lo g_{10} (y)$

Correct Answer:

Step-by-Step Solution

Insight: Taking the logarithm of both expressions reveals that their log values are identical due to the commutative property of multiplication.

Exam route: Apply $lo g_{10}$ to both $x$ and $y$ . Use the power rule to bring the exponents down. Compare the results.

Learning route:

Given $x = n^{l o g_{10} m}$ and $y = m^{l o g_{10} n}$ .

Take $lo g_{10}$ of both sides for $x$ :

$lo g_{10} x = lo g_{10} (n^{l o g_{10} m}) = (lo g_{10} m) \times (lo g_{10} n)$ .

Take $lo g_{10}$ of both sides for $y$ :

$lo g_{10} y = lo g_{10} (m^{l o g_{10} n}) = (lo g_{10} n) \times (lo g_{10} m)$ .

Since multiplication is commutative, $(lo g_{10} m) \times (lo g_{10} n) = (lo g_{10} n) \times (lo g_{10} m)$ .

Therefore, $lo g_{10} x = lo g_{10} y$ .

Since the logarithm function is one-to-one, this implies $x = y$ .

The relation is $x = y$ , matching option C.

Question 2 · Quantitative Aptitude · 2025 MCQ

If a real variable $x$ satisfies $3^{x^{2}} = 27 \times 9^{x}$ , then the value of $\frac{2 ^{x^{2}}}{( 2 ^{x} ) ^{2}}$ is:

A.
$2^{- 1}$
B.
$2^{0}$
C.
$2^{3}$
D.
$2^{15}$

Correct Answer:

Step-by-Step Solution

Insight: Unify the bases to 3 to find a relation between $x^{2}$ and $x$ , then substitute this relation directly into the target expression without solving for $x$ .

Exam route: Convert $27$ and $9^{x}$ to base 3. Equate exponents to get $x^{2} - 2 x = 3$ . Simplify the target expression to $2^{x^{2} - 2 x}$ and substitute 3.

Learning route:

Given $3^{x^{2}} = 27 \times 9^{x}$ .

Express all terms with base 3: $27 = 3^{3}$ and $9^{x} = (3^{2})^{x} = 3^{2 x}$ .

So, $3^{x^{2}} = 3^{3} \times 3^{2 x} = 3^{3 + 2 x}$ .

Since the bases are equal, equate the exponents:

$x^{2} = 3 + 2 x ⟹ x^{2} - 2 x = 3$ .

Now look at the target expression: $\frac{2 ^{x^{2}}}{( 2 ^{x} ) ^{2}}$ .

Using exponent laws, $(2^{x})^{2} = 2^{2 x}$ .

So the expression becomes $\frac{2 ^{x^{2}}}{2 ^{2 x}} = 2^{x^{2} - 2 x}$ .

Substitute $x^{2} - 2 x = 3$ into the exponent:

$2^{3} = 8$ .

The value is $2^{3}$ , which matches option C.

Question 3 · Quantitative Aptitude · 2024 MCQ

The sum of the following infinite series is

2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots

A.
$\frac{11}{3}$
B.
$\frac{7}{2}$
C.
$\frac{13}{4}$
D.
$\frac{9}{2}$

Correct Answer:

Step-by-Step Solution

Insight: The series is a mix of a constant, a geometric series with ratio 1/2, and another with ratio 1/3.

Exam route: Group the terms by their denominators' patterns. Sum the two infinite geometric series separately using $S = a / (1 - r)$ and add to the initial constant.

Learning route:

The given series is $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots$

Observe the denominators after the first term: 2, 4, 8, 16... are powers of 2. 3, 9, 27... are powers of 3.

We can split the series into three parts:

$S = 2 + (\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots) + (\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots)$

The first bracket is a geometric series with $a = 1/2$ and $r = 1/2$ . Its sum is $\frac{1/2}{1 - 1/2} = 1$ .

The second bracket is a geometric series with $a = 1/3$ and $r = 1/3$ . Its sum is $\frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = 1/2$ .

Total sum $S = 2 + 1 + 1/2 = 3.5 = 7/2$ .

Correct option is B.

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Notes written for recall, not for volume

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Unlock the whole course

Exponents, Logarithms and Infinite Series PYQs for GATE DA

Chapter Roadmap: Exponents, Logarithms and Infinite Series

Chapter Roadmap

1. Exponent and Logarithm Laws (Current)

2. Infinite Series Summation

What you will master:

The Logic of Exponents

The Logic of Exponents

Core Intuition: Operation Conversion

Exponents, Logarithms and Infinite Series: Solved Questions with Step-by-Step Explanations (3 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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