Logarithmic And Exponential Functions (Copy)
LOGARITHMIC AND EXPONENTIAL FUNCTIONS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Exponential Function | y = aˣ | Writing xᵃ |
| Exponential Growth | a > 1 | Confusing with decay |
| Exponential Decay | 0 < a < 1 | Confusing with growth |
| Logarithmic Function | y = logₐx | Forgetting x > 0 |
| Inverse Relationship | aˣ ↔ logₐx | Mixing inverse pairs |
| Log Definition | logₐx = y ↔ aʸ = x | Switching base and power |
| Log Multiplication Rule | logₐ(xy) = logₐx + logₐy | Multiplying logs |
| Log Division Rule | logₐ(x/y) = logₐx − logₐy | Dividing logs directly |
| Log Power Rule | logₐ(xⁿ) = nlogₐx | Forgetting power moves forward |
| Change of Base Formula | logₐx = logᵦx/logᵦa | Reversing numerator and denominator |
| Natural Logarithm | ln(x) = logₑx | Thinking ln is base 10 |
| Common Logarithm | log(x) = log₁₀x | Forgetting base 10 |
| Exponential Equation | Same bases → equate powers | Solving directly without same bases |
| Logarithmic Equation | Rewrite in exponential form | Keeping mixed forms |
| Domain Restriction | x > 0 for logs | Allowing negative values |
| Exponential Graph | Horizontal asymptote y = 0 | Drawing vertical asymptote |
| Logarithmic Graph | Vertical asymptote x = 0 | Drawing horizontal asymptote |
| Exponential Graph Shape | Increasing or decreasing curve | Drawing straight line |
| Logarithmic Graph Shape | Slow increasing curve | Wrong graph direction |
| Exponential Transformation | y = aˣ + k | Moving graph wrong direction |
| Reflection | y = a⁻ˣ | Forgetting reflection effect |
| Solving Logs | Convert to exponential form | Ignoring base |
| Solving Exponentials | Make bases equal | Decimal guessing |
| Exact Values | Leave exact forms when possible | Rounding too early |
| Laws of Indices | aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplying indices |
| Fractional Powers | a¹⁄ⁿ = ⁿ√a | Writing a/n |
| Negative Powers | a⁻ⁿ = 1/aⁿ | Leaving negative powers |
| Compound Growth | A = P(1 + r/100)ⁿ | Using simple interest formula |
| Exponential Decay Formula | A = P(1 − r/100)ⁿ | Wrong sign |
| Continuous Growth | A = Peⁿʳ | Forgetting e |
| Exponential Modelling | y = abˣ | Mixing a and b roles |
| Log Graph Intercept | Passes through (1,0) | Using (0,0) |
| Exponential Graph Intercept | Passes through (0,1) | Using (1,0) |
| Undefined Logs | logₐ0 undefined | Saying answer is 0 |
| Base Restriction | a > 0 and a ≠ 1 | Using invalid bases |
| Exponential Inequality | Solve using same bases/logs | Treating like linear equations |
| Logarithmic Inequality | Maintain domain restrictions | Ignoring x > 0 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK LOG RULES TABLE
| Rule | Formula |
|---|---|
| Multiplication | logₐ(xy) = logₐx + logₐy |
| Division | logₐ(x/y) = logₐx − logₐy |
| Power Rule | logₐ(xⁿ) = nlogₐx |
| Change of Base | logₐx = logᵦx/logᵦa |
| Log Definition | logₐx = y ↔ aʸ = x |
QUICK EXPONENTIAL RULES TABLE
| Rule | Formula |
|---|---|
| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Division | aᵐ/aⁿ = aᵐ⁻ⁿ |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ |
| Negative Power | a⁻ⁿ = 1/aⁿ |
| Fractional Power | a¹⁄ⁿ = ⁿ√a |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK GRAPH FEATURES TABLE
| Graph | Important Features |
|---|---|
| y = aˣ | Horizontal asymptote y = 0 |
| y = logₐx | Vertical asymptote x = 0 |
| Exponential Graph | Passes through (0,1) |
| Logarithmic Graph | Passes through (1,0) |
| Exponential Growth | Increasing rapidly |
| Exponential Decay | Decreasing gradually |
QUICK EQUATION SOLVING RULES
| Situation | Method |
|---|---|
| aˣ = aʸ | Equate powers |
| Different bases | Convert to same base |
| logₐx = b | Rewrite as aᵇ = x |
| logₐf(x) = logₐg(x) | Set f(x) = g(x) |
| Exponential inequality | Use same bases or logs |
COMMON EXACT VALUES
| Expression | Value |
|---|---|
| log₁₀1 | 0 |
| log₁₀10 | 1 |
| log₂8 | 3 |
| log₃27 | 3 |
| e⁰ | 1 |
| ln(1) | 0 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Exponential equations | Make bases equal |
| Log equations | Rewrite exponentially |
| Graph identification | Check asymptote direction |
| Domain questions | Logs require x > 0 |
| Growth/decay | Look at base value |
| Compound growth | Use repeated multiplier |
| Exact values | Keep in log/surd form |
| Change of base | Use calculator efficiently |
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Ignoring log restrictions | Invalid answers |
| Wrong log laws | Entire simplification incorrect |
| Mixing growth and decay | Wrong model |
| Incorrect graph asymptotes | Wrong sketch |
| Decimal rounding too early | Accuracy marks lost |
| Wrong base conversion | Entire equation incorrect |
| Leaving negative powers | Unsimplified answer |
| Using invalid log bases | Undefined mathematics |
| Forgetting inverse relationship | Wrong transformations |
| Graphing incorrectly | Feature marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.