Exponents, Logarithms and Infinite Series Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Exponents, Logarithms and Infinite Series notes for GATE DA: 18 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.

Standard Laws of Exponents

For real numbers $a, b > 0$ and rational numbers $m, n$

Product Rule

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

Quotient Rule

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

Power of Power

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

Negative Index

a^{- n} = \frac{1}{a ^{n}}

Zero Index

a^{0} = 1

Fractional Index

a^{1/ n} = n a

Distributive

(ab)^{n} = a^{n} b^{n}

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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Exponents, Logarithms and Infinite Series Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Exponents, Logarithms and Infinite Series notes for GATE DA: 18 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi

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