SL 1.5 — Laws of Exponents & Logarithms
| Focus Area | Meaning |
|---|---|
| Integer Exponents | Rules for multiplying, dividing, and raising powers with whole-number indices. |
| Logarithms (Base 10 & e) | Inverse operation of exponentiation, used to solve exponential equations. |
| Technology Use | Required for numerical evaluation of logarithms. |
📌 Laws of Exponents (Integer Powers)
Exponent laws simplify calculations involving powers and allow large expressions to be reduced logically.
- Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ
- Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a power: (aᵐ)ⁿ = aᵐⁿ
- Negative powers: a⁻ⁿ = 1 ÷ aⁿ
- Power of a product: (ab)ⁿ = aⁿbⁿ
Worked Examples:
5³ × 5⁻⁶ = 5⁻³ = 1 ÷ 125
6⁴ ÷ 6³ = 6¹ = 6
(2³)⁻⁴ = 2⁻¹²
(2x)⁴ = 16x⁴
2x⁻³ = 2 ÷ x³
🧠 Examiner Tip
Students often forget that negative powers mean reciprocals.
Always rewrite negative powers as fractions before final simplification.
Students often forget that negative powers mean reciprocals.
Always rewrite negative powers as fractions before final simplification.
🌍 Real-World Connection
Scientific notation, half-life decay, sound intensity, and light brightness all depend directly on exponent laws.
Scientific notation, half-life decay, sound intensity, and light brightness all depend directly on exponent laws.
📌 Introduction to Logarithms (Base 10 and Base e)
Logarithms are the inverse of exponentiation.
They answer the question: “What power do I raise the base to in order to get this number?”
- If aˣ = b, then logₐ(b) = x
- Base 10: log₁₀(x)
- Base e: ln(x)
- The argument b must always be > 0.
Meaning of ln(x): The natural logarithm uses base e ≈ 2.718 and appears in growth, decay, finance, and physics.
Examples:
log₁₀(1000) = 3 because 10³ = 1000
ln(e²) = 2
🔍 TOK Perspective
Is the number e discovered through nature or invented as a symbolic system?
Do logarithms exist independently of human definition?
Is the number e discovered through nature or invented as a symbolic system?
Do logarithms exist independently of human definition?
📌 Numerical Evaluation of Logarithms (Technology Required)
Exact values of most logarithms cannot be found manually and must be evaluated using calculators.
- log₁₀(2) ≈ 0.3010
- ln(5) ≈ 1.609
- log₁₀(0.01) = −2
📗 GDC Tip
Always confirm whether your calculator is using log (base 10) or
ln (base e).
Using the wrong base is a common exam mistake.
Always confirm whether your calculator is using log (base 10) or
ln (base e).
Using the wrong base is a common exam mistake.
📐 IA Spotlight
Strong IA themes include modelling sound levels, earthquakes, pH chemistry, population growth, or financial inflation using exponential and logarithmic functions.
Strong IA themes include modelling sound levels, earthquakes, pH chemistry, population growth, or financial inflation using exponential and logarithmic functions.
📌 Applications of Exponents & Logarithms
- Richter scale (earthquake intensity)
- Decibel scale (sound intensity)
- pH scale (acidity)
- Exponential population growth and radioactive decay
🌍 Real-World Connection
Every 1-unit increase on the Richter scale represents a 10× increase in earthquake strength, not a simple additive change.
Logarithms help compare events whose sizes differ by many orders of magnitude.
Every 1-unit increase on the Richter scale represents a 10× increase in earthquake strength, not a simple additive change.
Logarithms help compare events whose sizes differ by many orders of magnitude.
📝 Paper 1 Strategy
Whenever possible, simplify using exact exponent laws
before substituting numerical values.
This often earns method marks even if rounding errors occur later.
Whenever possible, simplify using exact exponent laws
before substituting numerical values.
This often earns method marks even if rounding errors occur later.