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Exponents, Logarithms and Infinite Series Notes for GATE DA

Exponents, Logarithms and Infinite Series notes for GATE DA: 13 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi

exponents logarithms and infinite series notes

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

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 \to Addition a^{m} \cdot a^{n} = a^{m + n} Division \to Subtraction \frac{a ^{m}}{a ^{n}} = a^{m - n} Powers \to Multiplication (a^{m})^{n} = a^{mn} Understanding this conversion is essential for simplifying complex algebraic structures in data science algorithms.

Logarithms as Inverse Exponents

Logarithms as Inverse Exponents The logarithm is defined as the inverse of exponentiation. lo g_{a} b = x ⟺ a^{x} = b Key Interpretation lo g_{a} b answers the question: "To what power must a be raised to produce b ?" Common Log lo g_{10} x (often lo g x) Natural Log lo g_{e} x (written ln x)

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Question 1

Level 1: Warm-up

For x > 0, let f (x) = x^{0} + x^{1} . What is the maximum value of f (x) on the interval [1, 3] ?

Question 2

Level 1: Warm-up

For a > 0, a \neq = 1 and x, y > 0, which of the following is always true?

Question 3

Level 1: Warm-up

For x > 0, let f (x) = 2 x^{1/2} - x^{0} . What is the maximum value of f (x) on the interval [4, 16] ?

Question 4

Level 1: Warm-up

For x > 0, which of the following expressions is equivalent to lo g_{3} (x^{2}) ?

Question 5

Level 1: Warm-up

For x \in [1, 16], let f (x) = 10 - x^{1/2} . What is the maximum value of f (x) ?

Question 6

Level 1: Warm-up

For x > 0 and x \neq = 1, consider the statements: P: lo g_{x} (x^{4}) = 4 Q: lo g_{x} (x + x^{3}) = 4 R: lo g_{x} (x^{2}) = 2 x Which of the following options is correct?

Question 7

Level 1: Warm-up

For x \in [1, 4], let f (x) = x^{- 1/2} . What is the maximum value of f (x) ?

Question 8

Level 1: Warm-up

For x > 0, which of the following expressions is equivalent to lo g_{5} (\frac{1}{x}) ?

Question 9

Level 1: Warm-up

For how many of the following infinite geometric series is it IMPOSSIBLE to compute a finite sum using the formula S = \frac{a}{1 - r} ? (P) 1 + \frac{1}{3} + \frac{1}{9} + \dots (Q) 2 + 4 + 8 + \dots (R) 5 - \frac{5}{2} + \frac{5}{4} - \dots (S) 1 + 1 + 1 + \dots

Question 10

Level 1: Warm-up

The sum of the infinite geometric series 1 + \frac{1}{4} + \frac{1}{16} + \dots is:

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Exponents, Logarithms and Infinite Series Notes for GATE DA

Exponents, Logarithms and Infinite Series notes for GATE DA: 13 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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.

Logarithms as Inverse Exponents

The logarithm is defined as the inverse of exponentiation.

lo g_{a} b = x ⟺ a^{x} = b

Key Interpretation

$lo g_{a} b$ answers the question: "To what power must $a$ be raised to produce $b$ ?"

Common Log

lo g_{10} x

(often

lo g x

)

Natural Log

lo g_{e} x

(written

ln x

)

Change of Base Formula

Use this to convert logarithms to a common base (usually 10 or $e$ ) when calculators or tables are limited.

lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}

Common Applications

loga​b=log10​alog10​b​

loga​b=lnalnb​

Use Case: Solving

2^{x} = 5 ⟹ x = \frac{l o g 5}{l o g 2}

Exponents, Logarithms and Infinite Series: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Quantitative Aptitude MCQ

For $x > 0$ , let $f (x) = x^{0} + x^{1}$ . What is the maximum value of $f (x)$ on the interval $[1, 3]$ ?

A.
4
B.
3
C.
2
D.
5

Correct Answer:

Step-by-Step Solution

Key idea: This tests the zero exponent rule ( $x^{0} = 1$ for $x > 0$ ) and evaluating a function on a closed interval.

Step 1: Simplify $f (x)$ . Since $x > 0$ , $x^{0} = 1$ . So $f (x) = 1 + x^{1} = 1 + x$ .

Step 2: Find the maximum on $[1, 3]$ . Since $f (x) = 1 + x$ is a linear increasing function, the maximum occurs at the right endpoint $x = 3$ .

Step 3: Evaluate $f (3) = 1 + 3 = 4$ .

Answer: 4

Common trap: Forgetting that $x^{0} = 1$ and treating it as $x^{0} = 0$ or $x^{0} = x$ , leading to wrong simplification.

Question 2 · Quantitative Aptitude MCQ

For $a > 0, a \neq = 1$ and $x, y > 0$ , which of the following is always true?

A.
$lo g_{a} (x + y) = lo g_{a} x + lo g_{a} y$
B.
$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
C.
$lo g_{a} (x - y) = lo g_{a} x - lo g_{a} y$
D.
$lo g_{a} (x / y) = lo g_{a} x / lo g_{a} y$

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fundamental logarithmic identities, specifically the product rule.

Step 1: Recall the product rule for logarithms: $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ for $x, y > 0$ .

Step 2: Check each option:

A: $lo g_{a} (x + y) \neq = lo g_{a} x + lo g_{a} y$ (log of sum is not sum of logs)
B: $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ ✓ (product rule)
C: $lo g_{a} (x - y) \neq = lo g_{a} x - lo g_{a} y$ (log of difference is not difference of logs)
D: $lo g_{a} (x / y) = lo g_{a} x - lo g_{a} y$ , not $lo g_{a} x / lo g_{a} y$ (quotient rule)

Answer: B

Common trap: Assuming log distributes over addition/subtraction like multiplication, or confusing the quotient rule.

Question 3 · Quantitative Aptitude MCQ

For $x > 0$ , let $f (x) = 2 x^{1/2} - x^{0}$ . What is the maximum value of $f (x)$ on the interval $[4, 16]$ ?

A.
7
B.
3
C.
8
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fractional exponent rule (roots) and the zero exponent rule, combined with finding the maximum of an increasing function on a closed interval.

Step 1: Simplify $f (x)$ . Since $x > 0$ , $x^{0} = 1$ . Also, $x^{1/2} = x$ .

So, $f (x) = 2 x - 1$ .

Step 2: Analyze the function on $[4, 16]$ . Since $x$ is strictly increasing for $x > 0$ , $f (x)$ is also strictly increasing. The maximum occurs at the right endpoint, $x = 16$ .

Step 3: Evaluate $f (16)$ .

$f (16) = 216 - 1 = 2 (4) - 1 = 8 - 1 = 7$ .

Answer: 7

Common trap: Forgetting the $- 1$ from $x^{0}$ and just calculating $216 = 8$ .

Question 4 · Quantitative Aptitude MCQ

For $x > 0$ , which of the following expressions is equivalent to $lo g_{3} (x^{2})$ ?

A.
$2 lo g_{3} x$
B.
$lo g_{3} (2 x)$
C.
$2 + lo g_{3} x$
D.
$(lo g_{3} x)^{2}$

Correct Answer:

Step-by-Step Solution

Key idea: This tests the power rule for logarithms.

Step 1: Recall the power rule: $lo g_{a} (x^{n}) = n lo g_{a} x$ .

Step 2: Apply the rule to the given expression.

$lo g_{3} (x^{2}) = 2 lo g_{3} x$ .

Step 3: Compare with the options. Option A matches exactly.

Answer: $2 lo g_{3} x$

Common trap: Confusing the power rule with the product rule, bringing the exponent down as an addition ( $2 + lo g_{3} x$ ) or squaring the entire log expression.

Question 5 · Quantitative Aptitude MCQ

For $x \in [1, 16]$ , let $f (x) = 10 - x^{1/2}$ . What is the maximum value of $f (x)$ ?

A.
6
B.
9
C.
10
D.
-6

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fractional exponent rule (roots) and finding the maximum of a decreasing function on a closed interval.

Step 1: Simplify $f (x)$ . The term $x^{1/2}$ is equivalent to $x$ .

So, $f (x) = 10 - x$ .

Step 2: Analyze the function on $[1, 16]$ . Since $x$ is strictly increasing for $x > 0$ , $- x$ is strictly decreasing. Therefore, $f (x)$ is strictly decreasing. The maximum occurs at the left endpoint, $x = 1$ .

Step 3: Evaluate $f (1)$ .

$f (1) = 10 - 1 = 10 - 1 = 9$ .

Answer: 9

Common trap: Evaluating at the right endpoint $x = 16$ instead of recognizing the function is decreasing, leading to $10 - 4 = 6$ .

Question 6 · Quantitative Aptitude MCQ

For $x > 0$ and $x \neq = 1$ , consider the statements:

P: $lo g_{x} (x^{4}) = 4$

Q: $lo g_{x} (x + x^{3}) = 4$

R: $lo g_{x} (x^{2}) = 2 x$

Which of the following options is correct?

A.
Only Q is true
B.
Only P is true
C.
Both P and R are true
D.
All are true

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fundamental logarithmic identities, specifically the power rule, and checks for common misconceptions about log distributing over addition.

Step 1: Evaluate Statement P.

Using the power rule $lo g_{a} (x^{n}) = n lo g_{a} x$ :

$lo g_{x} (x^{4}) = 4 lo g_{x} x = 4 (1) = 4$ .

So, P is true.

Step 2: Evaluate Statement Q.

The product rule applies to multiplication, not addition. $lo g_{x} (x + x^{3}) \neq = lo g_{x} x + lo g_{x} (x^{3})$ .

In fact, $lo g_{x} (x + x^{3}) = lo g_{x} (x (1 + x^{2})) = 1 + lo g_{x} (1 + x^{2})$ , which is not generally equal to 4.

So, Q is false.

Step 3: Evaluate Statement R.

Using the power rule: $lo g_{x} (x^{2}) = 2 lo g_{x} x = 2 (1) = 2$ .

The statement claims it is $2 x$ , which is false (unless $x = 1$ , but $x \neq = 1$ ).

So, R is false.

Answer: Only P is true.

Common trap: Assuming log distributes over addition, making Q true.

Question 7 · Quantitative Aptitude MCQ

For $x \in [1, 4]$ , let $f (x) = x^{- 1/2}$ . What is the maximum value of $f (x)$ ?

A.
1
B.
0.5
C.
2
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fractional and negative exponent rules, combined with finding the maximum of a decreasing function on a closed interval.

Step 1: Simplify $f (x)$ using exponent rules.

$x^{- 1/2} = \frac{1}{x ^{1/2}} = \frac{1}{x}$ .

Step 2: Analyze the function on $[1, 4]$ . Since $x$ is strictly increasing for $x > 0$ , its reciprocal $\frac{1}{x}$ is strictly decreasing.

Step 3: For a strictly decreasing function on a closed interval, the maximum value occurs at the left endpoint, $x = 1$ .

Step 4: Evaluate $f (1)$ .

$f (1) = \frac{1}{1} = 1$ .

Answer: 1

Common trap: Evaluating at the right endpoint $x = 4$ instead of recognizing the function is decreasing, leading to $1/ 4 = 0.5$ .

Question 8 · Quantitative Aptitude MCQ

For $x > 0$ , which of the following expressions is equivalent to $lo g_{5} (\frac{1}{x})$ ?

A.
$- lo g_{5} x$
B.
$\frac{1}{l o g _{5} x}$
C.
$lo g_{5} x - 1$
D.
$5 - lo g_{5} x$

Correct Answer:

Step-by-Step Solution

Key idea: This tests the reciprocal identity for logarithms, which is a direct consequence of the power rule.

Step 1: Rewrite the argument using a negative exponent.

$\frac{1}{x} = x^{- 1}$ .

Step 2: Apply the power rule for logarithms: $lo g_{a} (x^{n}) = n lo g_{a} x$ .

$lo g_{5} (x^{- 1}) = - 1 \cdot lo g_{5} x = - lo g_{5} x$ .

Step 3: Alternatively, use the quotient rule: $lo g_{a} (\frac{u}{v}) = lo g_{a} u - lo g_{a} v$ .

$lo g_{5} (\frac{1}{x}) = lo g_{5} 1 - lo g_{5} x = 0 - lo g_{5} x = - lo g_{5} x$ .

Answer: $- lo g_{5} x$

Common trap: Confusing the reciprocal of the argument with the reciprocal of the entire logarithm, leading to $\frac{1}{l o g _{5} x}$ .

Question 9 · Quantitative Aptitude MCQ

For how many of the following infinite geometric series is it IMPOSSIBLE to compute a finite sum using the formula $S = \frac{a}{1 - r}$ ?

(P) $1 + \frac{1}{3} + \frac{1}{9} + \dots$

(Q) $2 + 4 + 8 + \dots$

(R) $5 - \frac{5}{2} + \frac{5}{4} - \dots$

(S) $1 + 1 + 1 + \dots$

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: The geometric series formula $S = \frac{a}{1 - r}$ is only valid when $∣ r ∣ < 1$ . If $∣ r ∣ \geq 1$ , the series diverges and the formula gives nonsense.

Step 1: Check (P). Common ratio $r = \frac{1}{3}$ . Since $∣1/3∣ < 1$ , the series converges. Formula applies.

Step 2: Check (Q). Common ratio $r = 2$ . Since $∣2∣ > 1$ , the series diverges. Formula does NOT apply.

Step 3: Check (R). Common ratio $r = - \frac{1}{2}$ . Since $∣ - 1/2∣ < 1$ , the series converges. Formula applies.

Step 4: Check (S). Common ratio $r = 1$ . Since $∣1∣ = 1$ , the series diverges. Formula does NOT apply.

Series (Q) and (S) cannot have finite sums computed. Count = 2.

Answer: B

Question 10 · Quantitative Aptitude MCQ

The sum of the infinite geometric series $1 + \frac{1}{4} + \frac{1}{16} + \dots$ is:

A.
3/4
B.
4/3
C.
5/4
D.
4/5

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct application of the infinite geometric series sum formula.

Step 1: Identify the first term $a$ and the common ratio $r$ .

The series is $1 + 1/4 + 1/16 + \dots$

Here, the first term is $a = 1$ .

The common ratio is $r = \frac{1/4}{1} = \frac{1}{4}$ .

Step 2: Check for convergence.

Since $∣ r ∣ = 1/4 < 1$ , the series converges to a finite sum.

Step 3: Apply the geometric series sum formula.

The formula is $S = \frac{a}{1 - r}$ .

Substitute the values: $S = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3}$ .

Answer: B

exponents logarithms and infinite series notes

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

Logarithms as Inverse Exponents

Logarithms as Inverse Exponents

Key Interpretation

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Exponents, Logarithms and Infinite Series Notes 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

Logarithms as Inverse Exponents

Logarithms as Inverse Exponents

Key Interpretation

Change of Base Formula

Change of Base Formula

Common Applications

Exponents, Logarithms and Infinite Series: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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