What does log₂(8) = 3 mean?

It means 2³ = 8. In general, log_a(b) = c means a^c = b. The logarithm answers the question: 'To what power must I raise the base a to get b?'

What is the difference between log and ln?

log (without a base) usually means log base 10 (common logarithm). ln means log base e (natural logarithm, e ≈ 2.718). In aptitude exams, log almost always means log₁₀.

Why is log(0) undefined?

Because there is no real number x such that a^x = 0 for any positive base a. The function approaches negative infinity as the input approaches 0, but never equals 0.

Antilog is the inverse of log. If log₁₀(x) = n, then antilog(n) = x = 10ⁿ. For example, antilog(3) = 10³ = 1000.

Is log_a(b) × log_b(a) always 1?

Yes. Using the reciprocal property: log_a(b) = 1/log_b(a), so log_a(b) × log_b(a) = 1. This is a frequently tested exam fact.

Logarithm — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

Logarithm is the inverse of exponentiation.

If a^x = b, then log_a(b) = x

Read as: "log base a of b equals x"

Examples:

2³ = 8 → log₂(8) = 3
10² = 100 → log₁₀(100) = 2
5¹ = 5 → log₅(5) = 1
3⁰ = 1 → log₃(1) = 0

Key base facts:

Common log: base 10, written as log
Natural log: base e (≈ 2.718), written as ln
Binary log: base 2, written as log₂

All Log Properties

Property	Formula	Example
Product rule	log(mn) = log m + log n	log(6) = log 2 + log 3
Quotient rule	log(m/n) = log m − log n	log(5) = log 10 − log 2
Power rule	log(mⁿ) = n log m	log(8) = 3 log 2
Base of itself	log_a(a) = 1	log₅(5) = 1
Log of 1	log_a(1) = 0	log₁₀(1) = 0
Reciprocal	log_a(b) = 1/log_b(a)	log₂(8) = 1/log₈(2)
Change of base	log_a(b) = log_c(b)/log_c(a)	log₂(5) = log 5/log 2
Chain rule	log_a(b) × log_b(c) = log_a(c)	log₂(4) × log₄(8) = log₂(8)

Standard Values to Memorise

Expression	Value
log₁₀(1)	0
log₁₀(10)	1
log₁₀(100)	2
log₁₀(1000)	3
log₁₀(2)	0.3010
log₁₀(3)	0.4771
log₁₀(5)	0.6990 (= 1 − log 2)
log₁₀(7)	0.8451
log₁₀(e)	0.4343

Worked Examples

Example 1: Evaluate Using Definition

Problem: Find the value of log₃(243)

Step 1: Express 243 as a power of 3
  243 = 3⁵

Step 2: Apply definition
  log₃(3⁵) = 5 × log₃(3) = 5 × 1 = 5

Answer: log₃(243) = 5

Example 2: Apply Product Rule

Problem: Simplify log 4 + log 25

Step 1: Apply product rule
  log 4 + log 25 = log(4 × 25) = log(100)

Step 2: Evaluate
  log(100) = log(10²) = 2

Answer: log 4 + log 25 = 2

Note: This works because 4 × 25 = 100 = 10², a perfect power of 10.

Example 3: Quotient and Power Rules

Problem: If log 2 = 0.3010, find the value of log 32.

Step 1: Express 32 as a power of 2
  32 = 2⁵

Step 2: Apply power rule
  log 32 = log(2⁵) = 5 × log 2

Step 3: Substitute
  log 32 = 5 × 0.3010 = 1.5050

Answer: log 32 = 1.5050

Example 4: Combination of Properties

Problem: If log₁₀(2) = 0.3010 and log₁₀(3) = 0.4771,
find log₁₀(0.75).

Step 1: Express 0.75 as a fraction
  0.75 = 3/4 = 3/2²

Step 2: Apply quotient and power rules
  log(3/4) = log 3 − log 4
            = log 3 − log(2²)
            = log 3 − 2 log 2

Step 3: Substitute
  = 0.4771 − 2(0.3010)
  = 0.4771 − 0.6020
  = −0.1249

Answer: log₁₀(0.75) = −0.1249

Example 5: Solve for x (Log Equation)

Problem: Solve for x: log₂(x) + log₂(x−2) = 3

Step 1: Apply product rule
  log₂[x(x−2)] = 3

Step 2: Convert to exponential form
  x(x−2) = 2³ = 8

Step 3: Solve the quadratic
  x² − 2x − 8 = 0
  (x−4)(x+2) = 0
  x = 4 or x = −2

Step 4: Check validity (log requires positive argument)
  x = 4: log₂(4) + log₂(2) = 2 + 1 = 3 ✓
  x = −2: log₂(−2) is undefined ✗

Answer: x = 4

Example 6: Change of Base

Problem: Find log₈(512)

Method 1 — Definition:
  512 = 8³ (since 8¹=8, 8²=64, 8³=512)
  log₈(512) = 3

Method 2 — Change of base:
  log₈(512) = log 512 / log 8
             = log(2⁹) / log(2³)
             = 9 log 2 / (3 log 2)
             = 9/3
             = 3

Answer: log₈(512) = 3

Shortcuts and Tricks

Trick 1: The "Is it a power?" Check

Before applying any property, ask: "Can I write this as base^n?"

log₅(125) → 125 = 5³ → answer is 3
log₄(64) → 64 = 4³ → answer is 3

Trick 2: log + log = log ×

log a + log b = log(ab) — the plus becomes multiplication inside the log. log a − log b = log(a/b) — the minus becomes division inside the log.

Trick 3: The Reciprocal Trap

log_a(b) × log_b(a) = 1 (always)

This is asked directly in exams: "Find log₂(5) × log₅(2)" → Answer is 1.

Trick 4: Chain of Logs

log_a(b) × log_b(c) × log_c(d) = log_a(d)

Everything cancels in a chain except the outermost base and argument.

Trick 5: log(1) is Always Zero

No matter what base: log_a(1) = 0. This is a freebie in exams.

TNPSC Exam Pattern

Question Type	Frequency	Difficulty
Find log value using definition	Common	Easy
Simplify using product/quotient rule	Common	Easy–Medium
Find value given log 2 and log 3	Common	Medium
Solve log equation for x	Occasional	Medium
Reciprocal or chain log	Occasional	Medium

Practice Questions

Q1. Find log₇(343). Answer: 343 = 7³ → log₇(343) = 3

Q2. Simplify log 8 + log 125 (base 10). Answer: log(8 × 125) = log(1000) = 3

Q3. If log₁₀(2) = 0.3010, find log₁₀(50). Answer: log 50 = log(100/2) = log 100 − log 2 = 2 − 0.3010 = 1.6990

Q4. Find log₂(8) × log₈(32). Answer: log₂(32) = log₂(2⁵) = 5 (chain rule cancels base 8)

Q5. Solve: log₃(x) = 4. Answer: x = 3⁴ = 81

Q6. If log 2 = 0.3010 and log 3 = 0.4771, find log 12. Answer: log 12 = log(4×3) = log 4 + log 3 = 2(0.3010) + 0.4771 = 1.0791

Q7. Find x if log₅(x) + log₅(4) = log₅(100). Answer: log₅(4x) = log₅(100) → 4x = 100 → x = 25

Q8. What is log_a(a³)? Answer: 3 log_a(a) = 3 × 1 = 3

Q9. If log(0.0001) = x, find x. Answer: 0.0001 = 10⁻⁴ → log(10⁻⁴) = −4

Q10. Find the value of log₂(3) × log₃(4) × log₄(8). Answer: Chain rule → log₂(8) = log₂(2³) = 3