Summary
Exponentials and logarithms are mathematical concepts used to express and solve equations involving powers and their inverses. They are closely related through their properties and laws.
- Exponential Function (eˣ) — a function where the base is the constant e, approximately equal to 2.718. Example: e² is the exponential function with base e raised to the power of 2.
- Logarithm (log) — the inverse operation to exponentiation, indicating the power to which a base must be raised to obtain a number. Example: log₁₀(100) = 2 because 10² = 100.
- Natural Logarithm (ln) — a logarithm with base e, used to solve equations involving e. Example: ln(e³) = 3 because e³ is the power of e that gives e³.
- Laws of Logarithms — rules that simplify expressions: multiplication law, division law, and power law. Example: logₐ(xy) = logₐx + logₐy (multiplication law).
Exam Tips
Key Definitions to Remember
- Exponential Function: eˣ
- Logarithm: logₐx
- Natural Logarithm: ln(x)
Common Confusions
- Mixing up the base of logarithms
- Forgetting that logarithms are the inverse of exponentials
Typical Exam Questions
- What is log₁₀(100)? Answer: 2
- Solve the equation eˣ = 5. Answer: x = ln(5)
- Convert y = axⁿ to a linear form. Answer: ln(y) = ln(a) + nln(x)
What Examiners Usually Test
- Understanding of the relationship between exponentials and logarithms
- Ability to apply the laws of logarithms
- Solving equations and inequalities involving exponentials and logarithms
- Transforming non-linear relationships to linear form using logarithms