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Agniveer (All) Logarithms

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This page covers Agniveer (All) Logarithms with complete concept notes, 39 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Logarithms— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic
Product Rule

log_b(m x n) = log_b(m) + log_b(n) 2

Quotient Rule

log_b(m / n) = log_b(m) - log_b(n) 3

Power Rule

log_b(m^n) = n x log_b(m) 4

Base Change Rule

log_b(m) = log(m) / log(b) 5. log_b(b) = 1 (any base log of itself = 1) 6. log_b(1) = 0 (log of 1 is always zero, any base) 7. log_b(b^n) = n (direct simplification) 8. b^(log_b x) = x (inverse property) ---

2.Formula Block

Memorise — at least one formula appears in every paper
log_b(x) = y <=> b^y = x
log(mn) = log m + log n
log(m/n) = log m - log n
log(m^n) = n.log m
log_b(a) = 1 / log_a(b) [Reciprocal Rule]
log_b(a) = log_c(a) / log_c(b) [Base Change]

--

Note: log without base means log base 10 (common log). ln means log base e (natural log).

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3.Exam Patterns

What examiners ask — read before attempting PYQs

--- NDA asks logarithm questions in these common ways: • Simplify an expression using log properties • Find the value of a log expression (e.g., find x if log_2(x) = 5) • Prove or verify a log identity • Questions mixing base change with product/quotient rules • Word problems involving compound interest or population growth (log application) --- SHORTCUT / TRICK --- TRICK 1 — Reciprocal Flip: log_b(a) x log_a(b) = 1 So log_b(a) = 1 / log_a(b). If you see a product of two logs with flipped bases, the answer is 1. TRICK 2 — Chain Rule for Multiple Logs: log_a(b) x log_b(c) x log_c(d) = log_a(d) This chain cancels all middle terms. Very useful in NDA MCQs with 3-4 chained logs. ---

4.Worked Example

Solve this step-by-step before moving on
1
Step 1

Break each term using b^y = x. log_2(8) = log_2(2^3) = 3 log_2(4) = log_2(2^2) = 2 log_2(16) = log_2(2^4) = 4

2
Step 2

Substitute values. = 3 + 2 - 4 = 1 Answer: 1 Alternate check using Product/Quotient Rule: log_2(8) + log_2(4) - log_2(16) = log_2(8 x 4 / 16) = log_2(32 / 16) = log_2(2) = 1 ✓ ---

5.Exam Traps

Common mistakes students make — avoid these

--- Students write log(m + n) = log(m) + log(n). This is WRONG. Product rule applies to multiplication, NOT addition. log(m + n) cannot be simplified further. Remember: only log(m x n) splits into sum.

Key Points to Remember

  • log_b(x) = y means b^y = x — always convert to exponential form when confused
  • log of 1 is always 0 for any base: log_b(1) = 0
  • log of base itself is always 1: log_b(b) = 1
  • Product Rule: log(mn) = log m + log n — multiplication becomes addition
  • Power Rule: log(m^n) = n.log m — bring exponent down as multiplier
  • Base Change Formula: log_b(a) = log(a) / log(b) — use when bases differ
  • Reciprocal Rule: log_b(a) = 1 / log_a(b) — flip base and number, take reciprocal
  • Chain Rule: log_a(b) x log_b(c) = log_a(c) — middle terms cancel in chains

Exam-Specific Tips

  • log base 10 is called Common Logarithm; log base e is called Natural Logarithm (ln)
  • Value of log_10(2) = 0.3010 — memorise for quick calculation questions
  • Value of log_10(3) = 0.4771 — frequently used in simplification MCQs
  • log_10(e) = 0.4343 and ln(10) = 2.3026 — used in conversion between log types
  • For any base b: log_b(b^n) = n — direct answer without calculation
  • The base of a logarithm must be positive and not equal to 1; argument must be positive
  • log_a(b) x log_b(a) = 1 — product of reciprocal-base logs always equals 1
  • log_10(1000) = 3 because 10^3 = 1000 — standard benchmark value
Practice MCQs

Logarithms — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

If log₅(x) = 2, then x equals:

Practice 2easy

Solve for x: log₂(x + 1) = log₂(5) + log₂(3)

Practice 3easy

If log₂(x) + log₂(x - 2) = 3, then the value of x is:

Practice 4easy

Simplify: log₃(27) + log₃(9) - log₃(81)

Practice 5easy

If log₅(x) = 2 and log₅(y) = -1, then log₅(x/y) equals:

Practice 6easy

Solve for x: log₁₀(x + 1) = log₁₀(5) + log₁₀(2)

Practice 7easy

Simplify: log₃(27) + log₅(125) - log₂(8) = ?

Practice 8easy

If log₁₀(2) = 0.301 and log₁₀(3) = 0.477, then log₁₀(6) = ?

Practice 9easy

Solve for x: log₂(log₂(x)) = 1

Practice 10easy

If log_x(8) = 3/2, then x = ?

Practice 11easy

Simplify: log₃(27) + log₃(9) - log₃(3) = ?

Practice 12easy

If log₁₀(a) = 2 and log₁₀(b) = 3, then log₁₀(a/b) equals:

Practice 13easy

If log₂(log₃(x)) = 1, then x equals:

Practice 14medium

If log₂(x + 1) + log₂(x - 1) = 3, then x equals:

Practice 15medium

If log_x(64) = 2, then x equals:

Practice 16medium

If log₂(x² + 7x + 12) = 3, then the sum of all possible values of x is:

Practice 17medium

If log_a(b) · log_b(c) · log_c(a) = k, then k equals:

Practice 18medium

If log₂(log₃(log₅ x)) = 0, then the value of x is:

Practice 19medium

If log₁₀ 2 = 0.301 and log₁₀ 3 = 0.477, then log₁₀ 36 is equal to:

Practice 20medium

The number of solutions to the equation log₂(x - 1) + log₂(x + 2) = 3 is:

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60-Second Revision — Logarithms

  • Remember: log_b(x) = y is the same as b^y = x — switch freely between both forms
  • Formula: log(mn) = log m + log n AND log(m/n) = log m - log n — core rules
  • Formula: log(m^n) = n.log m — exponent comes down as a coefficient
  • Trick: log_a(b) x log_b(c) x log_c(d) = log_a(d) — chain logs cancel middle terms
  • Trap: log(m + n) is NOT equal to log m + log n — biggest error in exams
  • Remember: log_b(1) = 0 and log_b(b) = 1 — these are instant answers
  • Memorise: log 2 = 0.3010 and log 3 = 0.4771 — needed for numerical MCQs
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