Logarithm and Surds & Indices | General Aptitude for GATE, PSU, ESE, AE/JE

Logarithms, Surds, and Indices are fundamental topics in General Aptitude and are frequently asked in exams like GATE, PSU, ESE, AE/JE. A strong understanding of these concepts helps in solving numerical problems quickly and accurately.

📌 Indices (Exponents)

Indices represent the power of a number. If aⁿ is given, then a is the base and n is the exponent.

Important Laws of Indices

• a^m × aⁿ = a^m+n
• a^m / aⁿ = a^m−n
• (a^m)ⁿ = a^mn
• (ab)ⁿ = aⁿbⁿ
• a⁰ = 1
• a⁻ⁿ = 1 / aⁿ

Example:
2³ × 2⁴ = 2⁷ = 128

📌 Surds

A Surd is an irrational number expressed in root form, such as √2, √3, etc.

Types of Surds

• Pure Surd: √2, √5
• Mixed Surd: 2√3, 5√7
• Similar Surds: 2√3 and 5√3

Simplification of Surds

Example:
√50 = √(25 × 2) = 5√2

Rationalization

Rationalization is the process of removing surds from the denominator.

Example:
1 / √2 = √2 / 2

📌 Logarithms

Logarithm is the inverse operation of exponents.
If a^x = b, then log_ab = x

Important Logarithmic Laws

• log_a(mn) = log_am + log_an
• log_a(m/n) = log_am − log_an
• log_a(mⁿ) = n log_am
• log_aa = 1
• log_a1 = 0
• Change of base: log_ab = log_cb / log_ca

Example:
log₁₀1000 = 3 (because 10³ = 1000)

📊 Mixed Concept Example

Example:
Simplify: log₂(8) + √16 + 2³

log₂(8) = 3
√16 = 4
2³ = 8

Final Answer = 3 + 4 + 8 = 15

⚡ Key Tips for Exams

• Memorize all basic laws of indices and logarithms
• Simplify surds before calculation
• Use approximation techniques where needed
• Practice PYQs regularly for speed improvement

📝 Practice Questions

• Simplify: 3² × 3⁻¹
• Rationalize: 1 / (√3 + 1)
• Find: log₅(125)
• Simplify: √72
• If log_ab = 2, find log_ba

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